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

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 ....
View Full Document

## 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.

### Page1 / 4

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online