Induced connection on submanifold
For a submanifold of a Riemannian manifold
Suppose is a Riemannian manifold, is a submanifold. Then is a subbundle of the bundle . The Riemannian metric on naturally gives a notion of "projection" from to .
Then, given any linear connection on , we obtain an induced connection on . The induced connection is defined as follows. For , we have:
The definition has two key parts:
- We use a connection on to obtain a connection on . This can be done either by using the general notion of a pullback connection, or by the following more concrete process: First, using the tubular neighborhood theorem, consider a tubular neighborhood in . Extend to vector fields on by Fill this in later. Evaluate on this open set , to get a vector field on . Restrict the vector field to . This gives a section of the bundle
- We project this from to , using the Riemannian metric
Crudely speaking, we use the connection on , and project the output vector field we get, onto .