9c6c8fc5bd-p-77.pdf - 7.6 DISTANCIA EN EL CASO RIEMANNIANO...

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7.6. DISTANCIA EN EL CASO RIEMANNIANO 141 Se puede demostrar que no existen abiertos de la esfera y del plano que sean isom´ etricos entre s´ ı. Por otra parte, todo abierto simplemente conexo del cilindro es isom´ etrico a alg´un abierto del plano. En la de- terminaci´on de la posible existencia de isometr´ ıas desempe˜na un papel esencial el tensor de curvatura, que estudiaremos m´as adelante. 7.6. Distancia asociada a una m´ etrica rie- manniana Sea ( Q, g ) una variedad riemanniana y sea γ : [ a, b ] Q una curva diferenciable. Definimos la longitud de γ como L ( γ ) = b a γ ( t ) dt = b a g γ ( t ) ( γ ( t ) , γ ( t )) dt. El concepto de longitud de una curva que acabamos de introducir sugie- re la siguiente definici´on de distancia. Dada una variedad riemanniana conexa ( Q, g ) y dos puntos p, q Q , definimos la distancia entre ellos como: d g ( p, q ) = Inf γ { L ( γ ) : γ : [ a, b ] Q, γ ( a ) = p, γ ( b ) = q } . Aunque las propiedades de la funci´on distancia se ver´ an con m´as detalle en el Cap´ ıtulo ?? , anticipamos ahora las dos siguientes:
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  • Winter '15
  • Espacio vectorial, Cálculo tensorial, Producto escalar, Variedad de Riemann, Curvatura, Definici´on

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