En matemáticas , la geometría diferencial de superficies se ocupa de la geometría diferencial de superficies lisas con varias estructuras adicionales, la mayoría de las veces, una métrica de Riemann . Las superficies se han estudiado extensamente desde varias perspectivas: extrínsecamente , en relación con su incrustación en el espacio euclidiano e intrínsecamente , reflejando sus propiedades determinadas únicamente por la distancia dentro de la superficie medida a lo largo de las curvas en la superficie. Uno de los conceptos fundamentales investigados es la curvatura gaussiana , primero estudiado en profundidad porCarl Friedrich Gauss , [1] quien demostró que la curvatura era una propiedad intrínseca de una superficie, independiente de su incrustación isométrica en el espacio euclidiano.
Las superficies surgen naturalmente como gráficos de funciones de un par de variables y, a veces, aparecen en forma paramétrica o como loci asociados a curvas espaciales . Los grupos de Lie (en el espíritu del programa Erlangen ) han desempeñado un papel importante en su estudio , a saber, los grupos de simetría del plano euclidiano , la esfera y el plano hiperbólico . Estos grupos de Lie se pueden utilizar para describir superficies de curvatura gaussiana constante; también proporcionan un ingrediente esencial en el enfoque moderno de la geometría diferencial intrínseca a través de conexiones . Por otro lado, las propiedades extrínsecas que se basan en la incrustación de una superficie en el espacio euclidiano también se han estudiado extensamente. Esto está bien ilustrado por las ecuaciones no lineales de Euler-Lagrange en el cálculo de variaciones : aunque Euler desarrolló las ecuaciones de una variable para comprender las geodésicas , definidas independientemente de una incrustación, una de las principales aplicaciones de Lagrange de las dos ecuaciones variables fue a superficies mínimas , un concepto que solo se puede definir en términos de incrustación.
Historia
Arquímedes calculó los volúmenes de ciertas superficies cuádricas de revolución . [2] El desarrollo del cálculo en el siglo XVII proporcionó una forma más sistemática de calcularlos. [3] Euler estudió por primera vez la curvatura de las superficies generales . En 1760 [4] probó una fórmula para la curvatura de una sección plana de una superficie y en 1771 [5] consideró las superficies representadas en forma paramétrica. Monge sentó las bases de su teoría en sus memorias clásicas L'application de l'analyse à la géometrie que apareció en 1795. La contribución definitoria a la teoría de superficies fue hecha por Gauss en dos notables artículos escritos en 1825 y 1827. [ 1] Esto marcó un nuevo alejamiento de la tradición porque por primera vez Gauss consideró la geometría intrínseca de una superficie, las propiedades que están determinadas solo por las distancias geodésicas entre puntos en la superficie independientemente de la forma particular en que la superficie está ubicada en el espacio euclidiano ambiental. El resultado final , el Theorema Egregium de Gauss, estableció que la curvatura gaussiana es una invariante intrínseca, es decir, invariante bajo isometrías locales . Este punto de vista fue extendido a espacios de dimensiones superiores por Riemann y condujo a lo que hoy se conoce como geometría de Riemann . El siglo XIX fue la edad de oro para la teoría de superficies, tanto desde el punto de vista topológico como desde el punto de vista diferencial-geométrico, con la mayoría de los principales geómetras dedicándose a su estudio. [ cita requerida ] Darboux recopiló muchos resultados en su tratado de cuatro volúmenes Théorie des surface (1887-1896).
Descripción general
Intuitivamente es bastante familiar decir que la hoja de una planta, la superficie de un vaso o la forma de una cara, están curvadas de ciertas formas, y que todas estas formas, incluso después de ignorar cualquier marca distintiva, tienen ciertas formas geométricas. características que distinguen a unos de otros. La geometría diferencial de superficies se ocupa de la comprensión matemática de tales fenómenos. El estudio de este campo, que se inició en su forma moderna en la década de 1700, ha llevado al desarrollo de geometría abstracta y de dimensiones superiores, como la geometría riemanniana y la relatividad general . [ investigación original? ]
El objeto matemático esencial es el de una superficie regular. Aunque las convenciones varían en su definición precisa, estas forman una clase general de subconjuntos de espacio euclidiano tridimensional ( ℝ 3 ) que capturan parte de la noción familiar de "superficie". Al analizar la clase de curvas que se encuentran en dicha superficie y el grado en que las superficies las obligan a curvarse en ℝ 3 , se pueden asociar a cada punto de la superficie dos números, llamados curvaturas principales. Su promedio se llama curvatura media de la superficie y su producto se llama curvatura gaussiana.
Hay muchos ejemplos clásicos de superficies regulares, que incluyen:
- ejemplos familiares como planos, cilindros y esferas
- superficies mínimas , que se definen por la propiedad de que su curvatura media es cero en cada punto. Los ejemplos más conocidos son los catenoides y los helicoides , aunque se han descubierto muchos más. Las superficies mínimas también se pueden definir por propiedades relacionadas con el área de la superficie , con la consecuencia de que proporcionan un modelo matemático para la forma de las películas de jabón cuando se estiran sobre un marco de alambre.
- superficies regladas , que son superficies que tienen al menos una línea recta que atraviesa cada punto; los ejemplos incluyen el cilindro y el hiperboloide de una hoja.
Un resultado sorprendente de Carl Friedrich Gauss , conocido como el teorema egregium , mostró que la curvatura gaussiana de una superficie, que por su definición tiene que ver con cómo las curvas en la superficie cambian de dirección en el espacio tridimensional, en realidad se puede medir por las longitudes de curvas que se encuentran en las superficies junto con los ángulos que se forman cuando dos curvas en la superficie se cruzan. Terminológicamente, esto dice que la curvatura gaussiana se puede calcular a partir de la primera forma fundamental (también llamada tensor métrico ) de la superficie. La segunda forma fundamental , por el contrario, es un objeto que codifica cómo se distorsionan las longitudes y ángulos de las curvas en la superficie cuando las curvas se empujan fuera de la superficie.
A pesar de medir diferentes aspectos de longitud y ángulo, la primera y la segunda forma fundamental no son independientes entre sí y satisfacen ciertas restricciones llamadas ecuaciones de Gauss-Codazzi . Un teorema mayor, a menudo llamado teorema fundamental de la geometría diferencial de superficies, afirma que siempre que dos objetos satisfacen las restricciones de Gauss-Codazzi, surgirán como la primera y segunda formas fundamentales de una superficie regular.
Usando la primera forma fundamental, es posible definir nuevos objetos en una superficie regular. Las geodésicas son curvas en la superficie que satisfacen una cierta ecuación diferencial ordinaria de segundo orden que está especificada por la primera forma fundamental. Están muy directamente relacionados con el estudio de longitudes de curvas; una geodésica de longitud suficientemente corta será siempre la curva de menor longitud en la superficie que conecta sus dos extremos. Por lo tanto, las geodésicas son fundamentales para el problema de optimización de determinar el camino más corto entre dos puntos dados en una superficie regular.
También se puede definir el transporte paralelo a lo largo de cualquier curva dada, lo que da una receta sobre cómo deformar un vector tangente a la superficie en un punto de la curva a los vectores tangentes en todos los demás puntos de la curva. La prescripción está determinada por una ecuación diferencial ordinaria de primer orden que se especifica mediante la primera forma fundamental.
Los conceptos anteriores tienen esencialmente que ver con el cálculo multivariable. El teorema de Gauss-Bonnet es un resultado más global, que relaciona la curvatura gaussiana de una superficie junto con su tipo topológico. Afirma que el valor medio de la curvatura gaussiana está completamente determinado por la característica de Euler de la superficie junto con su área de superficie.
La noción de variedad de Riemann y superficie de Riemann son dos generalizaciones de las superficies regulares discutidas anteriormente. En particular, esencialmente toda la teoría de superficies regulares como se discute aquí tiene una generalización en la teoría de variedades de Riemann. Este no es el caso de las superficies de Riemann, aunque todas las superficies regulares dan un ejemplo de una superficie de Riemann.
Superficies regulares en el espacio euclidiano
Definición
Es intuitivamente claro que una esfera es lisa, mientras que un cono o una pirámide, debido a su vértice o bordes, no lo son. La noción de "superficie regular" es una formalización de la noción de superficie lisa. La definición utiliza la representación local de una superficie a través de mapas entre espacios euclidianos . Existe una noción estándar de suavidad para tales mapas; un mapa entre dos subconjuntos abiertos del espacio euclidiano es uniforme si sus derivadas parciales de cada orden existen en cada punto del dominio. [6] [7] [8]
A continuación se ofrecen tres formas equivalentes de presentar la definición; La definición del medio es quizás la más intuitiva visualmente, ya que esencialmente dice que una superficie regular es un subconjunto de ℝ 3 que es localmente la gráfica de una función suave (ya sea sobre una región en el plano yz , el plano xz o el xy avión).
Objetos utilizados en la definición | Una superficie regular en el espacio euclidiano ℝ 3 es un subconjunto S de ℝ 3 tal que cada punto de S tiene ... |
---|---|
Parametrizaciones locales | ... un vecindario abierto U ⊂ S para el cual hay un subconjunto abierto V de ℝ 2 y un homeomorfismo f : V → U tal que
|
Parches Monge | ... un vecindario abierto U ⊂ ℝ 3 para el cual hay un subconjunto abierto V de ℝ 2 y una función suave h : V → ℝ tal que uno de los siguientes se cumple:
|
Funciones implícitas | ... un vecindario abierto U ⊂ ℝ 3 para el que hay una función suave F : U → ℝ con:
|
Los homeomorfismos que aparecen en la primera definición se conocen como parametrizaciones locales o sistemas de coordenadas locales o cartas locales en S . [13] La equivalencia de las dos primeras definiciones afirma que, alrededor de cualquier punto de una superficie regular, siempre existen parametrizaciones locales de la forma ( u , v ) ↦ ( h ( u , v ), u , v ) , ( u , v ) ↦ ( u , h ( u , v ), v ) , o ( u , v ) ↦ ( u , v , h ( u , v )) , conocidos como parches de Monge. Las funciones F como en la tercera definición se denominan funciones de definición local . La equivalencia de las tres definiciones se deriva del teorema de la función implícita . [14] [15] [16]
Dadas dos parametrizaciones locales cualesquiera f : V → U y f ′: V ′ → U ′ de una superficie regular, la composición f −1 ∘ f ′ es necesariamente suave como un mapa entre subconjuntos abiertos de ℝ 2 . [17] Esto muestra que cualquier superficie regular tiene naturalmente la estructura de una variedad suave , con un atlas suave dado por las inversas de las parametrizaciones locales.
En la teoría clásica de la geometría diferencial, las superficies generalmente se estudian solo en el caso regular. [7] [18] Sin embargo, también es común estudiar superficies no regulares, en las que las dos derivadas parciales ∂ u f y ∂ v f de una parametrización local pueden no ser linealmente independientes . En este caso, S puede tener singularidades como bordes cúspides . Estas superficies se estudian típicamente en la teoría de la singularidad . Otras formas debilitadas de superficies regulares ocurren en el diseño asistido por computadora , donde una superficie se rompe en pedazos separados, con las derivadas de parametrizaciones locales que ni siquiera son continuas a lo largo de los límites. [ cita requerida ]
Ejemplos sencillos. Un ejemplo simple de una superficie regular viene dado por la 2-esfera {( x , y , z ) | x 2 + y 2 + z 2 = 1 }; esta superficie puede cubrirse con seis parches de Monge (dos de cada uno de los tres tipos indicados anteriormente), tomando h ( u , v ) = ± (1 - u 2 - v 2 ) 1/2 . También se puede cubrir mediante dos parametrizaciones locales, utilizando proyección estereográfica . El conjunto {( x , y , z ): (( x 2 + y 2 ) medio - r ) 2 + z 2 = R 2 } es un toro de revolución con radios r y R . Es una superficie regular; las parametrizaciones locales se pueden dar de la forma
El hiperboloide en dos hojas {( x , y , z ): z 2 = 1 + x 2 + y 2 } es una superficie regular; se puede cubrir con dos parches de Monge, con h ( u , v ) = ± (1 + u 2 + v 2 ) 1/2 . El helicoide aparece en la teoría de superficies mínimas . Está cubierto por una única parametrización local, f ( u , v ) = ( u sen v , u cos v , v ) .
Vectores tangentes y vectores normales
Deje que S sea una superficie regular en ℝ 3 , y dejar que p sea un elemento de S . Usando cualquiera de las definiciones anteriores, se pueden distinguir ciertos vectores en ℝ 3 como tangentes a S en p , y ciertos vectores en ℝ 3 como ortogonales a S en p .
Objetos utilizados en la definición | Un vector X en ℝ 3 es tangente a S en p si ... | Un vector n en ℝ 3 es normal a S en p si ... |
---|---|---|
Parametrizaciones locales | ... dada cualquier parametrización local f : V → S con p ∈ f ( V ) , X es una combinación lineal de y | ... es ortogonal a todo vector tangente a S en p |
Parches Monge | ... para cualquier parche de Monge ( u , v ) ↦ ( u , v , h ( u , v )) cuyo rango incluye p , uno tiene con las derivadas parciales evaluadas en el punto ( p 1 , p 2 ) . La definición análoga se aplica en el caso de los parches de Monge de las otras dos formas. | ... para cualquier parche de Monge ( u , v ) ↦ ( u , v , h ( u , v )) cuyo rango incluye p , n es un múltiplo de (∂ h/∂ u, ∂ h/∂ v, −1) evaluado en el punto ( p 1 , p 2 ) . La definición análoga se aplica en el caso de los parches de Monge de las otras dos formas. |
Funciones implícitas | ... para cualquier función de definición local F cuyo dominio contenga p , X es ortogonal a ∇ F ( p ) | ... para cualquier función de definición local F cuyo dominio contenga p , n es un múltiplo de ∇ F ( p ) |
Se ve que el espacio tangente o plano tangente a S en p , que se define como formado por todos los vectores tangentes a S en p , es un subespacio lineal bidimensional de ℝ 3 ; que a menudo se designa por T p S . El espacio normal a S en p , que se define para consistir en todos los vectores normales a S en p , es un subespacio lineal unidimensional de ℝ 3 que es ortogonal al espacio tangente T p S . Como tal, en cada punto p de S , hay dos vectores normales de longitud unitaria, llamados vectores normales unitarios. Es útil notar que los vectores normales unitarios en p se pueden dar en términos de parametrizaciones locales, parches de Monge o funciones de definición local, a través de las fórmulas
siguiendo las mismas notaciones que en las definiciones anteriores.
También es útil observar una definición "intrínseca" de vectores tangentes, que es típica de la generalización de la teoría de superficies regulares al establecimiento de variedades suaves . Define el espacio tangente como un espacio vectorial real bidimensional abstracto, en lugar de un subespacio lineal de ℝ 3 . En esta definición, se dice que un vector tangente a S en p es una asignación, a cada parametrización local f : V → S con p ∈ f ( V ) , de dos números X 1 y X 2 , tal que para cualquier otro local parametrización f ′: V → S con p ∈ f ( V ) (y con los números correspondientes ( X ′) 1 y ( X ′) 2 ), uno tiene
donde A f ′ ( p ) es la matriz jacobiana del mapeo f −1 ∘ f ′ , evaluada en el punto f ′ ( p ) . La colección de vectores tangentes a S en p tiene naturalmente la estructura de un espacio vectorial bidimensional. Un vector tangente en este sentido corresponde a un vector tangente en el sentido anterior considerando el vector
en ℝ 3 . La condición jacobiana en X 1 y X 2 asegura, por la regla de la cadena , que este vector no depende de f .
Para funciones uniformes en una superficie, los campos vectoriales (es decir, campos vectoriales tangentes) tienen una interpretación importante como operadores o derivaciones de primer orden. Dejar ser una superficie regular, un subconjunto abierto del plano y un gráfico de coordenadas. Si, el espacio se puede identificar con . similar identifica campos vectoriales en con campos vectoriales en . Tomando las variables estándar u y v , un campo vectorial tiene la forma, Con un y b funciones suaves. Si es un campo vectorial y es una función suave, entonces también es una función suave. El operador diferencial de primer ordenes una derivación , es decir, satisface la regla de Leibniz[19]
Para los campos vectoriales X e Y , es sencillo comprobar que el operadores una derivación correspondiente a un campo vectorial. Se llama corchete de mentira . Es simétrico sesgado y satisface la identidad de Jacobi:
En resumen, los campos vectoriales en o Forme un álgebra de Lie bajo el corchete de Lie. [20]
Primera y segunda formas fundamentales, el operador de forma y la curvatura
Sea S una superficie regular en ℝ 3 . Dada una parametrización local de f : V → S y un campo de vector normal unidad n a f ( V ) , se define los siguientes objetos como funciones de valores reales o de matriz con valores en V . La primera forma fundamental depende solo de f , y no de n . La cuarta columna registra la forma en que estas funciones dependen de f , relacionando las funciones E ′, F ′, G ′, L ′, etc., que surgen para una elección diferente de parametrización local, f ′: V ′ → S , a los que surjan por f . Aquí A denota la matriz jacobiana de f –1 ∘ f ′ . La relación clave para establecer las fórmulas de la cuarta columna es entonces
como sigue la regla de la cadena .
Terminología | Notación | Definición | Dependencia de la parametrización local |
---|---|---|---|
Primera forma fundamental | mi | ||
F | |||
GRAMO | |||
Segunda forma fundamental | L | ||
METRO | |||
norte | |||
Operador de forma [21] | PAG | ||
Curvatura gaussiana | K | ||
Curvatura media | H | ||
Curvaturas principales |
Mediante un cálculo directo con la matriz que define el operador de forma, se puede comprobar que la curvatura gaussiana es el determinante del operador de forma, la curvatura media es la traza del operador de forma y las curvaturas principales son los valores propios del operador de forma. ; además, la curvatura gaussiana es el producto de las curvaturas principales y la curvatura media es su suma. Estas observaciones también se pueden formular como definiciones de estos objetos. Estas observaciones también dejan claro que las últimas tres filas de la cuarta columna siguen inmediatamente a la fila anterior, ya que matrices similares tienen determinantes, trazas y valores propios idénticos. Es fundamental tener en cuenta que E , G y EG - F 2 son todos necesariamente positivos. Esto asegura que la matriz inversa en la definición del operador de forma esté bien definida y que las curvaturas principales sean números reales.
Tenga en cuenta también que una negación de la elección del campo vectorial normal unitario negará la segunda forma fundamental, el operador de forma, la curvatura media y las curvaturas principales, pero dejará la curvatura gaussiana sin cambios. En resumen, esto ha demostrado que, dada una superficie regular S , la curvatura gaussiana de S puede considerarse como una función de valor real en S ; en relación con una selección de campo vector normal unidad sobre todo de S , los dos curvaturas principales y la curvatura media son también valores reales-funciones en S .
Geométricamente, la primera y segunda formas fundamentales pueden ser vistos como dar información sobre la forma f ( u , v ) se mueve en ℝ 3 como ( u , v ) se mueve en V . En particular, la primera forma fundamental codifica la rapidez con que se mueve f , mientras que la segunda forma fundamental codifica la medida en que su movimiento es en la dirección del vector normal n . En otras palabras, la segunda forma fundamental en un punto p codifica la longitud de la proyección ortogonal desde S al plano tangente a S en p ; en particular, da la función cuadrática que mejor se aproxima a esta longitud. Este pensamiento se puede precisar mediante las fórmulas
como se desprende directamente de las definiciones de las formas fundamentales y del teorema de Taylor en dos dimensiones. Las principales curvaturas se pueden ver de la siguiente manera. En un punto dado P de S , considere la colección de todos los planos que contienen la línea ortogonal a S . Cada uno de esos planos tiene una curva de intersección con S , que puede considerarse como una curva plana dentro del propio plano. Las dos curvaturas principales en p son los valores máximos y mínimos posibles de la curvatura de esta curva plana en p , ya que el plano considerado gira alrededor de la línea normal.
A continuación se resume el cálculo de las cantidades anteriores relativas a un parche de Monge f ( u , v ) = ( u , v , h ( u , v )) . Aquí h u y h v denotan las dos derivadas parciales de h , con notación análoga para las segundas derivadas parciales. La segunda forma fundamental y todas las cantidades subsiguientes se calculan en relación con la elección dada de campo vectorial normal unitario.
Cantidad | Fórmula |
---|---|
Un campo vectorial normal unitario | |
Primera forma fundamental | |
Segunda forma fundamental | |
Operador de forma | |
Curvatura gaussiana | |
Curvatura media |
Símbolos de Christoffel, ecuaciones de Gauss-Codazzi y Theorema Egregium
Sea S una superficie regular en ℝ 3 . Los símbolos de Christoffel asignan, a cada parametrización local f : V → S , ocho funciones en V , definidas por [22]
También se pueden definir mediante las siguientes fórmulas, en las que n es un campo vectorial normal unitario a lo largo de f ( V ) y L , M , N son los componentes correspondientes de la segunda forma fundamental:
La clave de esta definición es que ∂ f/∂ u, ∂ f/∂ v, yn forman una base de ℝ 3 en cada punto, en relación con el cual cada una de las tres ecuaciones especifica de manera única los símbolos de Christoffel como coordenadas de las segundas derivadas parciales de f . La elección de la unidad normal no tiene ningún efecto sobre los símbolos de Christoffel, ya que si n se intercambia por su negación, entonces los componentes de la segunda forma fundamental también se niegan, por lo que los signos de Ln , Mn , Nn se dejan sin cambios.
La segunda definición muestra, en el contexto de las parametrizaciones locales, que los símbolos de Christoffel son geométricamente naturales. Aunque las fórmulas de la primera definición parecen menos naturales, tienen la importancia de mostrar que los símbolos de Christoffel pueden calcularse a partir de la primera forma fundamental, lo que no es inmediatamente evidente a partir de la segunda definición. La equivalencia de las definiciones se puede comprobar mediante la sustitución directamente la primera definición en el segundo, y el uso de las definiciones de E , F , G .
Las ecuaciones de Codazzi afirman que [23]
Estas ecuaciones pueden derivarse directamente de la segunda definición de los símbolos de Christoffel dada anteriormente; por ejemplo, la primera ecuación de Codazzi se obtiene diferenciando la primera ecuación con respecto av , la segunda ecuación con respecto a u , restando las dos y tomando el producto escalar con n . La ecuación de Gauss afirma que [24]
Estos pueden derivarse de manera similar a las ecuaciones de Codazzi, con uno utilizando las ecuaciones de Weingarten en lugar de tomar el producto escalar con n . Aunque se escriben como tres ecuaciones independientes, son idénticas cuando se sustituyen las definiciones de los símbolos de Christoffel, en términos de la primera forma fundamental. Hay muchas formas de escribir la expresión resultante, una de ellas derivada en 1852 por Brioschi utilizando un hábil uso de los determinantes: [25] [26]
Cuando se considera que los símbolos de Christoffel están definidos por la primera forma fundamental, las ecuaciones de Gauss y Codazzi representan ciertas restricciones entre la primera y la segunda forma fundamental. La ecuación de Gauss es particularmente digna de mención, ya que muestra que la curvatura gaussiana se puede calcular directamente a partir de la primera forma fundamental, sin necesidad de ninguna otra información; de manera equivalente, esto dice que LN - M 2 en realidad se puede escribir como una función de E , F , G , aunque los componentes individuales L , M , N no pueden. Esto se conoce como el teorema egregium y fue un gran descubrimiento de Carl Friedrich Gauss . Es particularmente sorprendente cuando uno recuerda la definición geométrica de la curvatura gaussiana de S como definida por los radios máximo y mínimo de círculos osculantes; parecen estar fundamentalmente definidos por la geometría de cómo S se dobla dentro de ℝ 3 . Sin embargo, el teorema muestra que su producto puede determinarse a partir de la geometría "intrínseca" de S , teniendo sólo que ver con las longitudes de las curvas a lo largo de S y los ángulos formados en sus intersecciones. Como dijo Marcel Berger : [27]
Este teorema es desconcertante. [...] Es el tipo de teorema que podría haber esperado decenas de años más antes de ser descubierto por otro matemático ya que, a diferencia de gran parte de la historia intelectual, no estaba en absoluto en el aire. [...] Hasta donde sabemos, no existe una prueba geométrica simple del teorema egregium en la actualidad.
Las ecuaciones de Gauss-Codazzi también se pueden expresar y derivar sucintamente en el lenguaje de las formas de conexión debido a Élie Cartan . [28] En el lenguaje del cálculo de tensores , haciendo uso de métricas naturales y conexiones en paquetes de tensores , la ecuación de Gauss se puede escribir como H 2 - | h | 2 = R y las dos ecuaciones de Codazzi se pueden escribir como ∇ 1 h 12 = ∇ 2 h 11 y ∇ 1 h 22 = ∇ 2 h 12 ; las expresiones complicado de hacer con símbolos de Christoffel y la primera forma fundamental están completamente absorbidos en las definiciones del tensor covariante derivado ∇ h y la curvatura escalar R . Pierre Bonnet demostró que dos formas cuadráticas que satisfacen las ecuaciones de Gauss-Codazzi siempre determinan de forma única una superficie incrustada localmente. [29] Por esta razón, las ecuaciones de Gauss-Codazzi a menudo se denominan ecuaciones fundamentales para superficies incrustadas, identificando con precisión de dónde provienen las curvaturas intrínseca y extrínseca. Admiten generalizaciones a superficies incrustadas en variedades riemannianas más generales .
Isometrías
Un difeomorfismo entre sets abiertos y en una superficie regular Se dice que es una isometría si conserva la métrica, es decir, la primera forma fundamental. [30] [31] [32] Así, para cada punto en y vectores tangentes a , hay igualdades
En términos del producto interno que proviene de la primera forma fundamental, esto se puede reescribir como
- .
Por otro lado, la longitud de una curva parametrizada se puede calcular como
y, si la curva se encuentra en , las reglas para el cambio de variables muestran que
Por el contrario, si preserva las longitudes de todos los parametrizados en curvas luego es una isometría. De hecho, para opciones adecuadas de, los vectores de la tangente y dar vectores tangentes arbitrarios y . Las igualdades deben ser válidas para todas las opciones de vectores tangentes. y así como y , así que eso . [33]
Un ejemplo sencillo de isometría lo proporcionan dos parametrizaciones. y de un set abierto en superficies regulares y . Si, y , luego es una isometría de sobre . [34]
El cilindro y el plano dan ejemplos de superficies que son localmente isométricas pero que no pueden extenderse a una isometría por razones topológicas. [35] Como otro ejemplo, el catenoide y el helicoide son localmente isométricos. [36]
Derivados covariantes
Un campo vectorial tangencial X en S asigna, a cada p en S , un vector tangente X p a S en p . De acuerdo con la definición "intrínseca" de los vectores tangentes dada anteriormente, un campo vectorial tangencial X asigna entonces, a cada parametrización local f : V → S , dos funciones de valor real X 1 y X 2 en V , de modo que
para cada p en S . Se dice que X es suave si las funciones X 1 y X 2 son suaves, para cualquier elección de f . [37] De acuerdo con las otras definiciones de vectores tangentes dadas anteriormente, también se puede considerar un campo vectorial tangencial X en S como un mapa X : S → ℝ 3 tal que X ( p ) está contenido en el espacio tangente T p S ⊂ ℝ 3 para cada p en S . Como es común en la situación más general de múltiples lisos , los campos de vectores tangenciales también se pueden definir como ciertos operadores diferenciales en el espacio de las funciones suaves en S .
Las derivadas covariantes (también llamadas "derivadas tangenciales") de Tullio Levi-Civita y Gregorio Ricci-Curbastro proporcionan un medio para diferenciar campos vectoriales tangenciales suaves. Dado un campo vectorial tangencial X y un vector tangente Y a S en p , la derivada covariante ∇ Y X es un cierto vector tangente a S en p . En consecuencia, si X e Y son campos vectoriales tangenciales, entonces ∇ Y X también puede considerarse como un campo vectorial tangencial; iterativamente, si X , Y y Z son campos vectoriales tangenciales, se puede calcular ∇ Z ∇ Y X , que será otro campo vectorial tangencial. Hay algunas formas de definir la derivada covariante; el primero a continuación utiliza los símbolos de Christoffel y la definición "intrínseca" de los vectores tangentes, y el segundo es más manifiestamente geométrico.
Dado un campo vectorial tangencial X y un vector tangente Y a S en p , uno define ∇ Y X que es el vector tangente a p que asigna a una parametrización local de f : V → S los dos números
donde D ( Y 1 , Y 2 ) es la derivada direccional . [38] Esto a menudo se abrevia en la forma menos engorrosa (∇ Y X ) k = ∂ Y ( X k ) + Y i Γ k
ijX j , haciendo uso de la notación de Einstein y entendiendo implícitamente las ubicaciones de la evaluación de funciones. Esto sigue una prescripción estándar en la geometría de Riemann para obtener una conexión a partir de una métrica de Riemann . Es un hecho fundamental que el vector
en ℝ 3 es independiente de la elección de la parametización local f , aunque esto es bastante tedioso de comprobar.
También se puede definir la derivada covariante mediante el siguiente enfoque geométrico, que no hace uso de símbolos de Christoffel o parametrizaciones locales. [39] [40] [41] Sea X un campo vectorial en S , visto como una función S → ℝ 3 . Dada cualquier curva c : ( a , b ) → S , se puede considerar la composición X ∘ c : ( a , b ) → ℝ 3 . Como mapa entre espacios euclidianos, se puede diferenciar en cualquier valor de entrada para obtener un elemento ( X ∘ c ) ′ ( t ) de ℝ 3 . La proyección ortogonal de este vector en T c ( t ) S define la covariante derivado ∇ c '( t ) X . Aunque esta es una definición muy limpia geométricamente, es necesario demostrar que el resultado sólo depende de c '( t ) y X , y no en C y X ; Las parametrizaciones locales se pueden utilizar para este pequeño argumento técnico.
It is not immediately apparent from the second definition that covariant differentiation depends only on the first fundamental form of S; however, this is immediate from the first definition, since the Christoffel symbols can be defined directly from the first fundamental form. It is straightforward to check that the two definitions are equivalent. The key is that when one regards X1∂f/∂u + X2∂f/∂v as a ℝ3-valued function, its differentiation along a curve results in second partial derivatives ∂2f; the Christoffel symbols enter with orthogonal projection to the tangent space, due to the formulation of the Christoffel symbols as the tangential components of the second derivatives of f relative to the basis ∂f/∂u, ∂f/∂v, n.[38] This is discussed in the above section.
The right-hand side of the three Gauss equations can be expressed using covariant differentiation. For instance, the right-hand side
can be recognized as the second coordinate of
relative to the basis ∂f/∂u, ∂f/∂v, as can be directly verified using the definition of covariant differentiation by Christoffel symbols. In the language of Riemannian geometry, this observation can also be phrased as saying that the right-hand sides of the Gauss equations are various components of the Ricci curvature of the Levi-Civita connection of the first fundamental form, when interpreted as a Riemannian metric.
Ejemplos de
Surfaces of revolution
A surface of revolution is obtained by rotating a curve in the xz-plane about the z-axis. Such surfaces include spheres, cylinders, cones, tori, and the catenoid. The general ellipsoids, hyperboloids, and paraboloids are not. Suppose that the curve is parametrized by
with s drawn from an interval (a, b). If c1 is never zero, if c1′ and c2′ are never both equal to zero, and if c1 and c2 are both smooth, then the corresponding surface of revolution
will be a regular surface in ℝ3. A local parametrization f : (a, b) × (0, 2π) → S is given by
Relative to this parametrization, the geometric data is:[42]
Quantity | Formula |
---|---|
A unit normal vector field | |
First fundamental form | |
Second fundamental form | |
Principal curvatures | |
Gaussian curvature | |
Mean curvature |
In the special case that the original curve is parametrized by arclength, i.e. (c1′(s))2 + (c1′(s))2 = 1, one can differentiate to find c1′(s)c1′′(s) + c2′(s)c2′′(s) = 0. On substitution into the Gaussian curvature, one has the simplified
The simplicity of this formula makes it particularly easy to study the class of rotationally symmetric surfaces with constant Gaussian curvature.[43] By reduction to the alternative case that c2(s) = s, one can study the rotationally symmetric minimal surfaces, with the result that any such surface is part of a plane or a scaled catenoid.[44]
Each constant-t curve on S can be parametrized as a geodesic; a constant-s curve on S can be parametrized as a geodesic if and only if c1′(s) is equal to zero. Generally, geodesics on S are governed by Clairaut's relation.
Quadric surfaces
Consider the quadric surface defined by[45]
This surface admits a parametrization
The Gaussian curvature and mean curvature are given by
Ruled surfaces
A ruled surface is one which can be generated by the motion of a straight line in E3.[46] Choosing a directrix on the surface, i.e. a smooth unit speed curve c(t) orthogonal to the straight lines, and then choosing u(t) to be unit vectors along the curve in the direction of the lines, the velocity vector v = ct and u satisfy
The surface consists of points
as s and t vary.
Then, if
the Gaussian and mean curvature are given by
The Gaussian curvature of the ruled surface vanishes if and only if ut and v are proportional,[47] This condition is equivalent to the surface being the envelope of the planes along the curve containing the tangent vector v and the orthogonal vector u, i.e. to the surface being developable along the curve.[48] More generally a surface in E3 has vanishing Gaussian curvature near a point if and only if it is developable near that point.[49] (An equivalent condition is given below in terms of the metric.)
Minimal surfaces
In 1760 Lagrange extended Euler's results on the calculus of variations involving integrals in one variable to two variables.[50] He had in mind the following problem:
Given a closed curve in E3, find a surface having the curve as boundary with minimal area.
Such a surface is called a minimal surface.
In 1776 Jean Baptiste Meusnier showed that the differential equation derived by Lagrange was equivalent to the vanishing of the mean curvature of the surface:
A surface is minimal if and only if its mean curvature vanishes.
Minimal surfaces have a simple interpretation in real life: they are the shape a soap film will assume if a wire frame shaped like the curve is dipped into a soap solution and then carefully lifted out. The question as to whether a minimal surface with given boundary exists is called Plateau's problem after the Belgian physicist Joseph Plateau who carried out experiments on soap films in the mid-nineteenth century. In 1930 Jesse Douglas and Tibor Radó gave an affirmative answer to Plateau's problem (Douglas was awarded one of the first Fields medals for this work in 1936).[51]
Many explicit examples of minimal surface are known explicitly, such as the catenoid, the helicoid, the Scherk surface and the Enneper surface. There has been extensive research in this area, summarised in Osserman (2002). In particular a result of Osserman shows that if a minimal surface is non-planar, then its image under the Gauss map is dense in S2.
Surfaces of constant Gaussian curvature
If a surface has constant Gaussian curvature, it is called a surface of constant curvature.[52]
- The unit sphere in E3 has constant Gaussian curvature +1.
- The Euclidean plane and the cylinder both have constant Gaussian curvature 0.
- The surfaces of revolution with φtt = φ have constant Gaussian curvature –1. Particular cases are obtained by taking φ(t) =C cosh t, C sinh t and C et.[53] The latter case is the classical pseudosphere generated by rotating a tractrix around a central axis. In 1868 Eugenio Beltrami showed that the geometry of the pseudosphere was directly related to that of the hyperbolic plane, discovered independently by Lobachevsky (1830) and Bolyai (1832). Already in 1840, F. Minding, a student of Gauss, had obtained trigonometric formulas for the pseudosphere identical to those for the hyperbolic plane.[54] The intrinsic geometry of this surface is now better understood in terms of the Poincaré metric on the upper half plane or the unit disc, and has been described by other models such as the Klein model or the hyperboloid model, obtained by considering the two-sheeted hyperboloid q(x, y, z) = −1 in three-dimensional Minkowski space, where q(x, y, z) = x2 + y2 – z2.[55]
Each of these surfaces of constant curvature has a transitive Lie group of symmetries. This group theoretic fact has far-reaching consequences, all the more remarkable because of the central role these special surfaces play in the geometry of surfaces, due to Poincaré's uniformization theorem (see below).
Other examples of surfaces with Gaussian curvature 0 include cones, tangent developables, and more generally any developable surface.
Estructura métrica local
For any surface embedded in Euclidean space of dimension 3 or higher, it is possible to measure the length of a curve on the surface, the angle between two curves and the area of a region on the surface. This structure is encoded infinitesimally in a Riemannian metric on the surface through line elements and area elements. Classically in the nineteenth and early twentieth centuries only surfaces embedded in R3 were considered and the metric was given as a 2×2 positive definite matrix varying smoothly from point to point in a local parametrization of the surface. The idea of local parametrization and change of coordinate was later formalized through the current abstract notion of a manifold, a topological space where the smooth structure is given by local charts on the manifold, exactly as the planet Earth is mapped by atlases today. Changes of coordinates between different charts of the same region are required to be smooth. Just as contour lines on real-life maps encode changes in elevation, taking into account local distortions of the Earth's surface to calculate true distances, so the Riemannian metric describes distances and areas "in the small" in each local chart. In each local chart a Riemannian metric is given by smoothly assigning a 2×2 positive definite matrix to each point; when a different chart is taken, the matrix is transformed according to the Jacobian matrix of the coordinate change. The manifold then has the structure of a 2-dimensional Riemannian manifold.
Shape operator
The differential dn of the Gauss map n can be used to define a type of extrinsic curvature, known as the shape operator[56] or Weingarten map. This operator first appeared implicitly in the work of Wilhelm Blaschke and later explicitly in a treatise by Burali-Forti and Burgati.[57] Since at each point x of the surface, the tangent space is an inner product space, the shape operator Sx can be defined as a linear operator on this space by the formula
for tangent vectors v, w (the inner product makes sense because dn(v) and w both lie in E3).[a] The right hand side is symmetric in v and w, so the shape operator is self-adjoint on the tangent space. The eigenvalues of Sx are just the principal curvatures k1 and k2 at x. In particular the determinant of the shape operator at a point is the Gaussian curvature, but it also contains other information, since the mean curvature is half the trace of the shape operator. The mean curvature is an extrinsic invariant. In intrinsic geometry, a cylinder is developable, meaning that every piece of it is intrinsically indistinguishable from a piece of a plane since its Gauss curvature vanishes identically. Its mean curvature is not zero, though; hence extrinsically it is different from a plane.
Equivalently, the shape operator can be defined as a linear operator on tangent spaces, Sp: TpM→TpM. If n is a unit normal field to M and v is a tangent vector then
(there is no standard agreement whether to use + or − in the definition).
In general, the eigenvectors and eigenvalues of the shape operator at each point determine the directions in which the surface bends at each point. The eigenvalues correspond to the principal curvatures of the surface and the eigenvectors are the corresponding principal directions. The principal directions specify the directions that a curve embedded in the surface must travel to have maximum and minimum curvature, these being given by the principal curvatures.
Curvas geodésicas en una superficie
Curves on a surface which minimize length between the endpoints are called geodesics; they are the shape that an elastic band stretched between the two points would take. Mathematically they are described using ordinary differential equations and the calculus of variations. The differential geometry of surfaces revolves around the study of geodesics. It is still an open question whether every Riemannian metric on a 2-dimensional local chart arises from an embedding in 3-dimensional Euclidean space: the theory of geodesics has been used to show this is true in the important case when the components of the metric are analytic.
Geodesics
Given a piecewise smooth path c(t) = (x(t), y(t)) in the chart for t in [a, b], its length is defined by
and energy by
The length is independent of the parametrization of a path. By the Euler–Lagrange equations, if c(t) is a path minimising length, parametrized by arclength, it must satisfy the Euler equations
where the Christoffel symbols Γk
ij are given by
where g11 = E, g12 = F, g22 = G and gij is the inverse matrix to gij. A path satisfying the Euler equations is called a geodesic. By the Cauchy–Schwarz inequality a path minimising energy is just a geodesic parametrised by arc length; and, for any geodesic, the parameter t is proportional to arclength.[58]
Geodesic curvature
The geodesic curvature kg at a point of a curve c(t), parametrised by arc length, on an oriented surface is defined to be[59]
where n(t) is the "principal" unit normal to the curve in the surface, constructed by rotating the unit tangent vector ċ(t) through an angle of +90°.
- The geodesic curvature at a point is an intrinsic invariant depending only on the metric near the point.
- A unit speed curve on a surface is a geodesic if and only if its geodesic curvature vanishes at all points on the curve.
- A unit speed curve c(t) in an embedded surface is a geodesic if and only if its acceleration vector c̈(t) is normal to the surface.
The geodesic curvature measures in a precise way how far a curve on the surface is from being a geodesic.
Orthogonal coordinates
When F = 0 throughout a coordinate chart, such as with the geodesic polar coordinates discussed below, the images of lines parallel to the x- and y-axes are orthogonal and provide orthogonal coordinates. If H = (EG)1⁄2, then the Gaussian curvature is given by[60]
If in addition E = 1, so that H = G1⁄2, then the angle φ at the intersection between geodesic (x(t),y(t)) and the line y = constant is given by the equation
The derivative of φ is given by a classical derivative formula of Gauss:[61]
Coordenadas polares geodésicas
Once a metric is given on a surface and a base point is fixed, there is a unique geodesic connecting the base point to each sufficiently nearby point. The direction of the geodesic at the base point and the distance uniquely determine the other endpoint. These two bits of data, a direction and a magnitude, thus determine a tangent vector at the base point. The map from tangent vectors to endpoints smoothly sweeps out a neighbourhood of the base point and defines what is called the "exponential map", defining a local coordinate chart at that base point. The neighbourhood swept out has similar properties to balls in Euclidean space, namely any two points in it are joined by a unique geodesic. This property is called "geodesic convexity" and the coordinates are called "normal coordinates". The explicit calculation of normal coordinates can be accomplished by considering the differential equation satisfied by geodesics. The convexity properties are consequences of Gauss's lemma and its generalisations. Roughly speaking this lemma states that geodesics starting at the base point must cut the spheres of fixed radius centred on the base point at right angles. Geodesic polar coordinates are obtained by combining the exponential map with polar coordinates on tangent vectors at the base point. The Gaussian curvature of the surface is then given by the second order deviation of the metric at the point from the Euclidean metric. In particular the Gaussian curvature is an invariant of the metric, Gauss's celebrated Theorema Egregium. A convenient way to understand the curvature comes from an ordinary differential equation, first considered by Gauss and later generalized by Jacobi, arising from the change of normal coordinates about two different points. The Gauss–Jacobi equation provides another way of computing the Gaussian curvature. Geometrically it explains what happens to geodesics from a fixed base point as the endpoint varies along a small curve segment through data recorded in the Jacobi field, a vector field along the geodesic.[62] One and a quarter centuries after Gauss and Jacobi, Marston Morse gave a more conceptual interpretation of the Jacobi field in terms of second derivatives of the energy function on the infinite-dimensional Hilbert manifold of paths.[63]
Exponential map
The theory of ordinary differential equations shows that if f(t, v) is smooth then the differential equation dv/dt = f(t, v) with initial condition v(0) = v0 has a unique solution for |t| sufficiently small and the solution depends smoothly on t and v0. This implies that for sufficiently small tangent vectors v at a given point p = (x0, y0), there is a geodesic cv(t) defined on (−2, 2) with cv(0) = (x0, y0) and ċv(0) = v. Moreover, if |s| ≤ 1, then csv = cv(st). The exponential map is defined by
- expp(v) = cv (1)
and gives a diffeomorphism between a disc ‖v‖ < δ and a neighbourhood of p; more generally the map sending (p, v) to expp(v) gives a local diffeomorphism onto a neighbourhood of (p, p). The exponential map gives geodesic normal coordinates near p.[64]
Computation of normal coordinates
There is a standard technique (see for example Berger (2004)) for computing the change of variables to normal coordinates u, v at a point as a formal Taylor series expansion. If the coordinates x, y at (0,0) are locally orthogonal, write
- x(u,v) = αu + L(u,v) + λ(u,v) + …
- y(u,v) = βv + M(u,v) + μ(u,v) + …
where L, M are quadratic and λ, μ cubic homogeneous polynomials in u and v. If u and v are fixed, x(t) = x(tu,tv) and y(t) = y(tu, tv) can be considered as formal power series solutions of the Euler equations: this uniquely determines α, β, L, M, λ and μ.
Gauss's lemma
In these coordinates the matrix g(x) satisfies g(0) = I and the lines t ↦ tv are geodesics through 0. Euler's equations imply the matrix equation
- g(v)v = v,
a key result, usually called the Gauss lemma. Geometrically it states that
the geodesics through 0 cut the circles centred at 0 orthogonally.
Taking polar coordinates (r,θ), it follows that the metric has the form
- ds2 = dr2 + G(r,θ) dθ2.
In geodesic coordinates, it is easy to check that the geodesics through zero minimize length. The topology on the Riemannian manifold is then given by a distance function d(p,q), namely the infimum of the lengths of piecewise smooth paths between p and q. This distance is realised locally by geodesics, so that in normal coordinates d(0,v) = ‖v‖. If the radius δ is taken small enough, a slight sharpening of the Gauss lemma shows that the image U of the disc ‖v‖ < δ under the exponential map is geodesically convex, i.e. any two points in U are joined by a unique geodesic lying entirely inside U.[65][66]
Theorema Egregium
Gauss's Theorema Egregium, the "Remarkable Theorem", shows that the Gaussian curvature of a surface can be computed solely in terms of the metric and is thus an intrinsic invariant of the surface, independent of any isometric embedding in E3 and unchanged under coordinate transformations. In particular isometries of surfaces preserve Gaussian curvature.[67]
This theorem can expressed in terms of the power series expansion of the metric, ds, is given in normal coordinates (u, v) as
- ds2 = du2 + dv2 − K(u dv – v du)2/12 + ….
Gauss–Jacobi equation
Taking a coordinate change from normal coordinates at p to normal coordinates at a nearby point q, yields the Sturm–Liouville equation satisfied by H(r,θ) = G(r,θ)1⁄2, discovered by Gauss and later generalised by Jacobi,
Hrr = –KH
The Jacobian of this coordinate change at q is equal to Hr. This gives another way of establishing the intrinsic nature of Gaussian curvature. Because H(r,θ) can be interpreted as the length of the line element in the θ direction, the Gauss–Jacobi equation shows that the Gaussian curvature measures the spreading of geodesics on a geometric surface as they move away from a point.[68]
Laplace–Beltrami operator
On a surface with local metric
and Laplace–Beltrami operator
where H2 = EG − F2, the Gaussian curvature at a point is given by the formula[69]
where r denotes the geodesic distance from the point.
In isothermal coordinates, first considered by Gauss, the metric is required to be of the special form
In this case the Laplace–Beltrami operator is given by
and φ satisfies Liouville's equation[70]
Isothermal coordinates are known to exist in a neighbourhood of any point on the surface, although all proofs to date rely on non-trivial results on partial differential equations.[71] There is an elementary proof for minimal surfaces.[72]
Teorema de Gauss-Bonnet
On a sphere or a hyperboloid, the area of a geodesic triangle, i.e. a triangle all the sides of which are geodesics, is proportional to the difference of the sum of the interior angles and π. The constant of proportionality is just the Gaussian curvature, a constant for these surfaces. For the torus, the difference is zero, reflecting the fact that its Gaussian curvature is zero. These are standard results in spherical, hyperbolic and high school trigonometry (see below). Gauss generalised these results to an arbitrary surface by showing that the integral of the Gaussian curvature over the interior of a geodesic triangle is also equal to this angle difference or excess. His formula showed that the Gaussian curvature could be calculated near a point as the limit of area over angle excess for geodesic triangles shrinking to the point. Since any closed surface can be decomposed up into geodesic triangles, the formula could also be used to compute the integral of the curvature over the whole surface. As a special case of what is now called the Gauss–Bonnet theorem, Gauss proved that this integral was remarkably always 2π times an integer, a topological invariant of the surface called the Euler characteristic. This invariant is easy to compute combinatorially in terms of the number of vertices, edges, and faces of the triangles in the decomposition, also called a triangulation. This interaction between analysis and topology was the forerunner of many later results in geometry, culminating in the Atiyah-Singer index theorem. In particular properties of the curvature impose restrictions on the topology of the surface.
Geodesic triangles
Gauss proved that, if Δ is a geodesic triangle on a surface with angles α, β and γ at vertices A, B and C, then
In fact taking geodesic polar coordinates with origin A and AB, AC the radii at polar angles 0 and α:
where the second equality follows from the Gauss–Jacobi equation and the fourth from Gauss' derivative formula in the orthogonal coordinates (r,θ).
Gauss' formula shows that the curvature at a point can be calculated as the limit of angle excess α + β + γ − π over area for successively smaller geodesic triangles near the point. Qualitatively a surface is positively or negatively curved according to the sign of the angle excess for arbitrarily small geodesic triangles.[49]
Gauss–Bonnet theorem
Since every compact oriented 2-manifold M can be triangulated by small geodesic triangles, it follows that
where χ(M) denotes the Euler characteristic of the surface.
In fact if there are F faces, E edges and V vertices, then 3F = 2E and the left hand side equals 2πV – πF = 2π(V – E + F) = 2πχ(M).
This is the celebrated Gauss–Bonnet theorem: it shows that the integral of the Gaussian curvature is a topological invariant of the manifold, namely the Euler characteristic. This theorem can be interpreted in many ways; perhaps one of the most far-reaching has been as the index theorem for an elliptic differential operator on M, one of the simplest cases of the Atiyah-Singer index theorem. Another related result, which can be proved using the Gauss–Bonnet theorem, is the Poincaré-Hopf index theorem for vector fields on M which vanish at only a finite number of points: the sum of the indices at these points equals the Euler characteristic, where the index of a point is defined as follows: on a small circle round each isolated zero, the vector field defines a map into the unit circle; the index is just the winding number of this map.)[49][73][74]
Curvature and embeddings
If the Gaussian curvature of a surface M is everywhere positive, then the Euler characteristic is positive so M is homeomorphic (and therefore diffeomorphic) to S2. If in addition the surface is isometrically embedded in E3, the Gauss map provides an explicit diffeomorphism. As Hadamard observed, in this case the surface is convex; this criterion for convexity can be viewed as a 2-dimensional generalisation of the well-known second derivative criterion for convexity of plane curves. Hilbert proved that every isometrically embedded closed surface must have a point of positive curvature. Thus a closed Riemannian 2-manifold of non-positive curvature can never be embedded isometrically in E3; however, as Adriano Garsia showed using the Beltrami equation for quasiconformal mappings, this is always possible for some conformally equivalent metric.[75]
Superficies de curvatura constante
The simply connected surfaces of constant curvature 0, +1 and –1 are the Euclidean plane, the unit sphere in E3, and the hyperbolic plane. Each of these has a transitive three-dimensional Lie group of orientation preserving isometries G, which can be used to study their geometry. Each of the two non-compact surfaces can be identified with the quotient G / K where K is a maximal compact subgroup of G. Here K is isomorphic to SO(2). Any other closed Riemannian 2-manifold M of constant Gaussian curvature, after scaling the metric by a constant factor if necessary, will have one of these three surfaces as its universal covering space. In the orientable case, the fundamental group Γ of M can be identified with a torsion-free uniform subgroup of G and M can then be identified with the double coset space Γ \ G / K. In the case of the sphere and the Euclidean plane, the only possible examples are the sphere itself and tori obtained as quotients of R2 by discrete rank 2 subgroups. For closed surfaces of genus g ≥ 2, the moduli space of Riemann surfaces obtained as Γ varies over all such subgroups, has real dimension 6g − 6.[76] By Poincaré's uniformization theorem, any orientable closed 2-manifold is conformally equivalent to a surface of constant curvature 0, +1 or –1. In other words, by multiplying the metric by a positive scaling factor, the Gaussian curvature can be made to take exactly one of these values (the sign of the Euler characteristic of M).[77]
Euclidean geometry
In the case of the Euclidean plane, the symmetry group is the Euclidean motion group, the semidirect product of the two dimensional group of translations by the group of rotations.[78] Geodesics are straight lines and the geometry is encoded in the elementary formulas of trigonometry, such as the cosine rule for a triangle with sides a, b, c and angles α, β, γ:
Flat tori can be obtained by taking the quotient of R2 by a lattice, i.e. a free Abelian subgroup of rank 2. These closed surfaces have no isometric embeddings in E3. They do nevertheless admit isometric embeddings in E4; in the easiest case this follows from the fact that the torus is a product of two circles and each circle can be isometrically embedded in E2.[79]
Spherical geometry
The isometry group of the unit sphere S2 in E3 is the orthogonal group O(3), with the rotation group SO(3) as the subgroup of isometries preserving orientation. It is the direct product of SO(3) with the antipodal map, sending x to –x.[80] The group SO(3) acts transitively on S2. The stabilizer subgroup of the unit vector (0,0,1) can be identified with SO(2), so that S2 = SO(3)/SO(2).
The geodesics between two points on the sphere are the great circle arcs with these given endpoints. If the points are not antipodal, there is a unique shortest geodesic between the points. The geodesics can also be described group theoretically: each geodesic through the North pole (0,0,1) is the orbit of the subgroup of rotations about an axis through antipodal points on the equator.
A spherical triangle is a geodesic triangle on the sphere. It is defined by points A, B, C on the sphere with sides BC, CA, AB formed from great circle arcs of length less than π. If the lengths of the sides are a, b, c and the angles between the sides α, β, γ, then the spherical cosine law states that
The area of the triangle is given by
- Area = α + β + γ − π.
Using stereographic projection from the North pole, the sphere can be identified with the extended complex plane C ∪ {∞}. The explicit map is given by
Under this correspondence every rotation of S2 corresponds to a Möbius transformation in SU(2), unique up to sign.[81] With respect to the coordinates (u, v) in the complex plane, the spherical metric becomes[82]
The unit sphere is the unique closed orientable surface with constant curvature +1. The quotient SO(3)/O(2) can be identified with the real projective plane. It is non-orientable and can be described as the quotient of S2 by the antipodal map (multiplication by −1). The sphere is simply connected, while the real projective plane has fundamental group Z2. The finite subgroups of SO(3), corresponding to the finite subgroups of O(2) and the symmetry groups of the platonic solids, do not act freely on S2, so the corresponding quotients are not 2-manifolds, just orbifolds.
Hyperbolic geometry
Non-Euclidean geometry[83] was first discussed in letters of Gauss, who made extensive computations at the turn of the nineteenth century which, although privately circulated, he decided not to put into print. In 1830 Lobachevsky and independently in 1832 Bolyai, the son of one Gauss' correspondents, published synthetic versions of this new geometry, for which they were severely criticized. However it was not until 1868 that Beltrami, followed by Klein in 1871 and Poincaré in 1882, gave concrete analytic models for what Klein dubbed hyperbolic geometry. The four models of 2-dimensional hyperbolic geometry that emerged were:
- the Beltrami-Klein model;
- the Poincaré disk;
- the Poincaré upper half-plane;
- the hyperboloid model of Wilhelm Killing in 3-dimensional Minkowski space.
The first model, based on a disk, has the advantage that geodesics are actually line segments (that is, intersections of Euclidean lines with the open unit disk). The last model has the advantage that it gives a construction which is completely parallel to that of the unit sphere in 3-dimensional Euclidean space. Because of their application in complex analysis and geometry, however, the models of Poincaré are the most widely used: they are interchangeable thanks to the Möbius transformations between the disk and the upper half-plane.
Let
be the Poincaré disk in the complex plane with Poincaré metric
In polar coordinates (r, θ) the metric is given by
The length of a curve γ:[a,b] → D is given by the formula
The group G = SU(1,1) given by
acts transitively by Möbius transformations on D and the stabilizer subgroup of 0 is the rotation group
The quotient group SU(1,1)/±I is the group of orientation-preserving isometries of D. Any two points z, w in D are joined by a unique geodesic, given by the portion of the circle or straight line passing through z and w and orthogonal to the boundary circle. The distance between z and w is given by
In particular d(0,r) = 2 tanh−1 r and c(t) = 1/2tanh t is the geodesic through 0 along the real axis, parametrized by arclength.
The topology defined by this metric is equivalent to the usual Euclidean topology, although as a metric space (D,d) is complete.
A hyperbolic triangle is a geodesic triangle for this metric: any three points in D are vertices of a hyperbolic triangle. If the sides have length a, b, c with corresponding angles α, β, γ, then the hyperbolic cosine rule states that
The area of the hyperbolic triangle is given by[84]
- Area = π – α – β – γ.
The unit disk and the upper half-plane
are conformally equivalent by the Möbius transformations
Under this correspondence the action of SL(2,R) by Möbius transformations on H corresponds to that of SU(1,1) on D. The metric on H becomes
Since lines or circles are preserved under Möbius transformations, geodesics are again described by lines or circles orthogonal to the real axis.
The unit disk with the Poincaré metric is the unique simply connected oriented 2-dimensional Riemannian manifold with constant curvature −1. Any oriented closed surface M with this property has D as its universal covering space. Its fundamental group can be identified with a torsion-free concompact subgroup Γ of SU(1,1), in such a way that
In this case Γ is a finitely presented group. The generators and relations are encoded in a geodesically convex fundamental geodesic polygon in D (or H) corresponding geometrically to closed geodesics on M.
Examples.
- the Bolza surface of genus 2;
- the Klein quartic of genus 3;
- the Macbeath surface of genus 7;
- the First Hurwitz triplet of genus 14.
Uniformization
Given an oriented closed surface M with Gaussian curvature K, the metric on M can be changed conformally by scaling it by a factor e2u. The new Gaussian curvature K′ is then given by
where Δ is the Laplacian for the original metric. Thus to show that a given surface is conformally equivalent to a metric with constant curvature K′ it suffices to solve the following variant of Liouville's equation:
When M has Euler characteristic 0, so is diffeomorphic to a torus, K′ = 0, so this amounts to solving
By standard elliptic theory, this is possible because the integral of K over M is zero, by the Gauss–Bonnet theorem.[85]
When M has negative Euler characteristic, K′ = −1, so the equation to be solved is:
Using the continuity of the exponential map on Sobolev space due to Neil Trudinger, this non-linear equation can always be solved.[86]
Finally in the case of the 2-sphere, K′ = 1 and the equation becomes:
So far this non-linear equation has not been analysed directly, although classical results such as the Riemann-Roch theorem imply that it always has a solution.[87] The method of Ricci flow, developed by Richard S. Hamilton, gives another proof of existence based on non-linear partial differential equations to prove existence.[88] In fact the Ricci flow on conformal metrics on S2 is defined on functions u(x, t) by
After finite time, Chow showed that K′ becomes positive; previous results of Hamilton could then be used to show that K′ converges to +1.[89] Prior to these results on Ricci flow, Osgood, Phillips & Sarnak (1988) had given an alternative and technically simpler approach to uniformization based on the flow on Riemannian metrics g defined by log det Δg.
A simple proof using only elliptic operators discovered in 1988 can be found in Ding (2001). Let G be the Green's function on S2 satisfying ΔG = 1 + 4πδP, where δP is the point measure at a fixed point P of S2. The equation Δv = 2K – 2, has a smooth solution v, because the right hand side has integral 0 by the Gauss–Bonnet theorem. Thus φ = 2G + v satisfies Δφ = 2K away from P. It follows that g1 = eφg is a complete metric of constant curvature 0 on the complement of P, which is therefore isometric to the plane. Composing with stereographic projection, it follows that there is a smooth function u such that e2ug has Gaussian curvature +1 on the complement of P. The function u automatically extends to a smooth function on the whole of S2.[b]
Conexión riemanniana y transporte paralelo
The classical approach of Gauss to the differential geometry of surfaces was the standard elementary approach[90] which predated the emergence of the concepts of Riemannian manifold initiated by Bernhard Riemann in the mid-nineteenth century and of connection developed by Tullio Levi-Civita, Élie Cartan and Hermann Weyl in the early twentieth century. The notion of connection, covariant derivative and parallel transport gave a more conceptual and uniform way of understanding curvature, which not only allowed generalisations to higher dimensional manifolds but also provided an important tool for defining new geometric invariants, called characteristic classes.[91] The approach using covariant derivatives and connections is nowadays the one adopted in more advanced textbooks.[92]
Covariant derivative
Connections on a surface can be defined from various equivalent but equally important points of view. The Riemannian connection or Levi-Civita connection.[93] is perhaps most easily understood in terms of lifting vector fields, considered as first order differential operators acting on functions on the manifold, to differential operators on the tangent bundle or frame bundle. In the case of an embedded surface, the lift to an operator on vector fields, called the covariant derivative, is very simply described in terms of orthogonal projection. Indeed, a vector field on a surface embedded in R3 can be regarded as a function from the surface into R3. Another vector field acts as a differential operator component-wise. The resulting vector field will not be tangent to the surface, but this can be corrected taking its orthogonal projection onto the tangent space at each point of the surface. As Ricci and Levi-Civita realised at the turn of the twentieth century, this process depends only on the metric and can be locally expressed in terms of the Christoffel symbols.
Parallel transport
Parallel transport of tangent vectors along a curve in the surface was the next major advance in the subject, due to Levi-Civita.[49] It is related to the earlier notion of covariant derivative, because it is the monodromy of the ordinary differential equation on the curve defined by the covariant derivative with respect to the velocity vector of the curve. Parallel transport along geodesics, the "straight lines" of the surface, can also easily be described directly. A vector in the tangent plane is transported along a geodesic as the unique vector field with constant length and making a constant angle with the velocity vector of the geodesic. For a general curve, this process has to be modified using the geodesic curvature, which measures how far the curve departs from being a geodesic.[65]
A vector field v(t) along a unit speed curve c(t), with geodesic curvature kg(t), is said to be parallel along the curve if
- it has constant length
- the angle θ(t) that it makes with the velocity vector ċ(t) satisfies
This recaptures the rule for parallel transport along a geodesic or piecewise geodesic curve, because in that case kg = 0, so that the angle θ(t) should remain constant on any geodesic segment. The existence of parallel transport follows because θ(t) can be computed as the integral of the geodesic curvature. Since it therefore depends continuously on the L2 norm of kg, it follows that parallel transport for an arbitrary curve can be obtained as the limit of the parallel transport on approximating piecewise geodesic curves.[94]
The connection can thus be described in terms of lifting paths in the manifold to paths in the tangent or orthonormal frame bundle, thus formalising the classical theory of the "moving frame", favoured by French authors.[95] Lifts of loops about a point give rise to the holonomy group at that point. The Gaussian curvature at a point can be recovered from parallel transport around increasingly small loops at the point. Equivalently curvature can be calculated directly at an infinitesimal level in terms of Lie brackets of lifted vector fields.
Connection 1-form
The approach of Cartan and Weyl, using connection 1-forms on the frame bundle of M, gives a third way to understand the Riemannian connection. They noticed that parallel transport dictates that a path in the surface be lifted to a path in the frame bundle so that its tangent vectors lie in a special subspace of codimension one in the three-dimensional tangent space of the frame bundle. The projection onto this subspace is defined by a differential 1-form on the orthonormal frame bundle, the connection form. This enabled the curvature properties of the surface to be encoded in differential forms on the frame bundle and formulas involving their exterior derivatives.
This approach is particularly simple for an embedded surface. Thanks to a result of Kobayashi (1956), the connection 1-form on a surface embedded in Euclidean space E3 is just the pullback under the Gauss map of the connection 1-form on S2.[96] Using the identification of S2 with the homogeneous space SO(3)/SO(2), the connection 1-form is just a component of the Maurer–Cartan 1-form on SO(3).[97]
Geometría diferencial global de superficies
Although the characterisation of curvature involves only the local geometry of a surface, there are important global aspects such as the Gauss–Bonnet theorem, the uniformization theorem, the von Mangoldt-Hadamard theorem, and the embeddability theorem. There are other important aspects of the global geometry of surfaces.[98] These include:
- Injectivity radius, defined as the largest r such that two points at a distance less than r are joined by a unique geodesic. Wilhelm Klingenberg proved in 1959 that the injectivity radius of a closed surface is bounded below by the minimum of δ = π/√sup K and the length of its smallest closed geodesic. This improved a theorem of Bonnet who showed in 1855 that the diameter of a closed surface of positive Gaussian curvature is always bounded above by δ; in other words a geodesic realising the metric distance between two points cannot have length greater than δ.
- Rigidity. In 1927 Cohn-Vossen proved that two ovaloids – closed surfaces with positive Gaussian curvature – that are isometric are necessarily congruent by an isometry of E3. Moreover, a closed embedded surface with positive Gaussian curvature and constant mean curvature is necessarily a sphere; likewise a closed embedded surface of constant Gaussian curvature must be a sphere (Liebmann 1899). Heinz Hopf showed in 1950 that a closed embedded surface with constant mean curvature and genus 0, i.e. homeomorphic to a sphere, is necessarily a sphere; five years later Alexandrov removed the topological assumption. In the 1980s, Wente constructed immersed tori of constant mean curvature in Euclidean 3-space.
- Carathéodory conjecture: This conjecture states that a closed convex three times differentiable surface admits at least two umbilic points. The first work on this conjecture was in 1924 by Hans Hamburger, who noted that it follows from the following stronger claim: the half-integer valued index of the principal curvature foliation of an isolated umbilic is at most one.
- Zero Gaussian curvature: a complete surface in E3 with zero Gaussian curvature must be a cylinder or a plane.
- Hilbert's theorem (1901): no complete surface with constant negative curvature can be immersed isometrically in E3.
- The Willmore conjecture. This conjecture states that the integral of the square of the mean curvature of a torus immersed in E3 should be bounded below by 2π2. It is known that the integral is Moebius invariant. It was solved in 2012 by Fernando Codá Marques and André Neves.[99]
- Isoperimetric inequalities. In 1939 Schmidt proved that the classical isoperimetric inequality for curves in the Euclidean plane is also valid on the sphere or in the hyperbolic plane: namely he showed that among all closed curves bounding a domain of fixed area, the perimeter is minimized by when the curve is a circle for the metric. In one dimension higher, it is known that among all closed surfaces in E3 arising as the boundary of a bounded domain of unit volume, the surface area is minimized for a Euclidean ball.
- Systolic inequalities for curves on surfaces. Given a closed surface, its systole is defined to be the smallest length of any non-contractible closed curve on the surface. In 1949 Loewner proved a torus inequality for metrics on the torus, namely that the area of the torus over the square of its systole is bounded below by √3/2, with equality in the flat (constant curvature) case. A similar result is given by Pu's inequality for the real projective plane from 1952, with a lower bound of 2/π also attained in the constant curvature case. For the Klein bottle, Blatter and Bavard later obtained a lower bound of √8/π. For a closed surface of genus g, Hebda and Burago showed that the ratio is bounded below by 1/2. Three years later Mikhail Gromov found a lower bound given by a constant times g1⁄2, although this is not optimal. Asymptotically sharp upper and lower bounds given by constant times g/(log g)2 are due to Gromov and Buser-Sarnak, and can be found in Katz (2007). There is also a version for metrics on the sphere, taking for the systole the length of the smallest closed geodesic. Gromov conjectured a lower bound of 1/2√3 in 1980: the best result so far is the lower bound of 1/8 obtained by Regina Rotman in 2006.[100]
Guía de lectura
One of the most comprehensive introductory surveys of the subject, charting the historical development from before Gauss to modern times, is by Berger (2004). Accounts of the classical theory are given in Eisenhart (2004), Kreyszig (1991) and Struik (1988); the more modern copiously illustrated undergraduate textbooks by Gray, Abbena & Salamon (2006), Pressley (2001) and Wilson (2008) might be found more accessible. An accessible account of the classical theory can be found in Hilbert & Cohn-Vossen (1952). More sophisticated graduate-level treatments using the Riemannian connection on a surface can be found in Singer & Thorpe (1967), do Carmo (2016) and O'Neill (2006).
Ver también
- Flatness (mathematics)
- Tangent vector
- Zoll surface
Notas
- ^ Note that in some more recent texts the symmetric bilinear form on the right hand side is referred to as the second fundamental form; however, it does not in general correspond to the classically defined second fundamental form.
- ^ This follows by an argument involving a theorem of Sacks & Uhlenbeck (1981) on removable singularities of harmonic maps of finite energy.
- ^ a b Gauss 1902.
- ^ Struik 1987, pp. 50–53
- ^ Wells 2017, pp. 17–30
- ^ Euler 1760
- ^ Euler 1771
- ^ Kreysig 1991
- ^ a b Struik 1988
- ^ Warner 1983
- ^ Hitchin 2013, p. 45
- ^ do Carmo 2016, pp. 54–56
- ^ Wilson 2008, p. 115
- ^ Pressley, pp. 68–77
- ^ do Carmo 2016, pp. 55
- ^ do Carmo 2016, pp. 60–65
- ^ O'Neill 2006, p. 113
- ^ Lee "Introduction to Smooth Manifolds"
- ^ do Carmo 2016, pp. 72
- ^ Kreyszig 1991
- ^ Singer & Thorpe 1966, p. 100–114
- ^ Singer & Thorpe 1966, p. 133–134
- ^ Do Carmo 2016, pp. 155–157
- ^ Do Carmo, page 235
- ^ Do Carmo, page 238
- ^ Do Carmo, pages 237-238
- ^ Struik 1961, p. 112
- ^ Darboux, Vol. III, page 246
- ^ Berger. A panoramic view of Riemannian geometry.
- ^ O'Neill 2006, p. 257
- ^ do Carmo 2016, pp. 309–314
- ^ Hitchin 2013, p. 57
- ^ do Carmo 2016, p. 221–222
- ^ O'Neill 2006, pp. 281–289
- ^ Hitchin 2013, pp. 57–58
- ^ do Carmo 2016, p. 223
- ^ do Carmo 2016, pp. 222–223
- ^ do Carmo 2016, pp. 224–225
- ^ Do Carmo, page 183
- ^ a b Do Carmo, page 242
- ^ Hitchin 2013
- ^ Struik 1961
- ^ O'Neill 2006
- ^ Spivak, "A comprehensive introduction to differential geometry, vol. III." Page 157.
- ^ Spivak, pages 161-166
- ^ Spivak, page 168
- ^ Eisenhart 2004, pp. 228–229
- ^ Eisenhart 2004, pp. 241–250; do Carmo 2016, pp. 188–197.
- ^ do Carmo 2016, p. 194.
- ^ Eisenhart 2004, pp. 61–65.
- ^ a b c d Eisenhart 2004
- ^ Eisenhart 2004, pp. 250–269; do Carmo 2016, pp. 197–213.
- ^ Douglas' solution is described in Courant (1950).
- ^ Eisenhart 2004, pp. 270–291; O'Neill, pp. 249–251 ; Hilbert & Cohn-Vossen 1952.
- ^ O'Neill, pp. 249–251 ; do Carmo, pp. 168–170 ; Gray, Abbena & Salamon 2006.
- ^ Stillwell 1996, pp. 1–5.
- ^ Wilson 2008.
- ^ O'Neill 2006, pp. 195–216; do Carmo 2016, pp. 134–153; Singer & Thorpe 1967, pp. 216–224.
- ^ Gray, Abbena & Salamon 2006, p. 386.
- ^ Berger 2004; Wilson 2008; Milnor 1963 .
- ^ Eisenhart 2002, p. 131 ; Berger 2004, p. 39; do Carmo 2016, p. 248; O'Neill 2006, p. 237
- ^ Eisenhart 2004; Taylor 1996a, pp. 472–473, Appendix C.
- ^ Eisenhart 2004, section 88; Berger 2004.
- ^ do Carmo 2016, p. 357
- ^ Milnor 1963
- ^ Wilson 2008
- ^ a b Berger 2004
- ^ do Carmo 2016, pp. 303–305
- ^ Berger 2004, pp. 41, 61, 123–124
- ^ O'Neill 2006, p. 395
- ^ Helgason 1978, p. 92
- ^ O'Neill 2006, p. 286
- ^ do Carmo 2016, p. 227
- ^ Osserman 2002, pp. 31–32
- ^ do Carmo 2016, pp. 283–286
- ^ Thorpe 1994, pp. 201–207
- ^ Singer & Thorpe 1967; Garsia, Adriano M. (1961), "An imbedding of closed Riemann surfaces in Euclidean space", Comment. Math. Helv., 35: 93–110, doi:10.1007/BF02567009
- ^ Imayoshi & Taniguchi 1992, pp. 47–49
- ^ Berger 1977; Taylor 1996 .
- ^ Wilson 2008, pp. 1–23, Chapter I, Euclidean geometry.
- ^ do Carmo 2016.
- ^ Wilson 2008, pp. 25–49, Chapter II, Spherical geometry.
- ^ Wilson 2008, Chapter 2.
- ^ Eisenhart 2004, p. 110.
- ^ Stillwell 1990 ; Bonola, Carslaw & Enriques 1955.
- ^ Wilson 2008, Chapter 5.
- ^ Taylor 1996b, p. 107; Berger 1977, pp. 341–343.
- ^ Berger 1977, pp. 222–225; Taylor 1996b, pp. 101–108.
- ^ Taylor 1996b
- ^ Chow 1991
- ^ Chen, Lu & Tian (2006) pointed out and corrected a missing step in the approach of Hamilton and Chow; see also Andrews & Bryan (2009) .
- ^ Eisenhart 2004; Kreyszig 1991; Berger 2004; Wilson 2008.
- ^ Kobayashi & Nomizu 1969, Chapter XII.
- ^ do Carmo 2016; O'Neill 2006; Singer & Thorpe 1967.
- ^ Levi-Civita 1917
- ^ Arnold 1989, pp. 301–306, Appendix I.; Berger 2004, pp. 263–264.
- ^ Darboux.
- ^ Kobayashi & Nomizu 1969
- ^ Ivey & Landsberg 2003.
- ^ Berger 2004, pp. 145–161; do Carmo 2016; Chern 1967; Hopf 1989.
- ^ Codá Marques, Fernando; Neves, André (2014). "Min-Max theory and the Willmore conjecture". Annals of Mathematics. 179 (2): 683–782. arXiv:1202.6036. doi:10.4007/annals.2014.179.2.6. JSTOR 24522767.
- ^ Rotman, R. (2006) "The length of a shortest closed geodesic and the area of a 2-dimensional sphere", Proc. Amer. Math. Soc. 134: 3041-3047. Previous lower bounds had been obtained by Croke, Rotman-Nabutovsky and Sabourau.
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enlaces externos
- Media related to Differential geometry of surfaces at Wikimedia Commons