riem - Riemannian Connections 1 General Definition of a...

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Unformatted text preview: Riemannian Connections 1. General Definition of a Riemannian Connection. Let ( M, g ) be a smooth manifold with a smooth Riemannian metric g . Let V be the set of all smooth vector fields on M . A connection or a covariant derivative on M is an operator ∇ : V × V → V , that, using the notation ∇ ( X, Y ) = ∇ X Y , satisfies the following axioms: (1) ∇ X 1 + X 2 Y = ∇ X 1 Y + ∇ X 2 Y (2) ∇ X ( Y 1 + Y 2 ) = ∇ X Y 1 + ∇ X Y 2 (3) ∇ fX Y = f ∇ X Y , where f : M → R is any smooth function. (4) ∇ X ( fY ) = X ( f ) Y + f ∇ X Y , where f : M → R is any smooth function. A connection ∇ on M is called a Riemannian connection with respect to g if in addition to the above axioms, ∇ also satisfies the following two axioms: (5) X ( g ( Y, Z ) ) = g ( ∇ X Y, Z ) + g ( Y, ∇ X Z ) for any X, Y, Z ∈ V . (6) ∇ X Y- ∇ Y X = [ X, Y ]. Theorem. For any smooth manifold M with a smooth Riemann- ian metric g there exists a unique Riemannian connection ∇ on M corresponding to g . 2. The case of R n . Let M = R n with the standard inner product giving on R n . Then ∇ X Y := D X Y is a Riemannian connection on M . Here D X Y is the directional derivative of Y with respect to X : D X Y | p := lim t → Y ( p + tX )- Y ( p ) t = lim t → Y ( c ( t ))- Y ( p ) t where c ( t ) is any smooth curve with c (0) = p and ˙ c (0) = X ( p ). Let c ( t ) be a curve and Y ( t ) be a vector field along c , that is for every t Y ( t ) is a tangent vector to R 3 at the point c ( t ). Then the directional derivative of Y with respect to c is: D ˙ c Y ( t ) = lim s → Y ( t + s )- Y ( t ) s ....
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This note was uploaded on 04/23/2010 for the course MATH Math 481 taught by Professor Kapovich during the Fall '08 term at University of Illinois at Urbana–Champaign.

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riem - Riemannian Connections 1 General Definition of a...

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