Induced connection on submanifold
From Diffgeom
Definition
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
.