Induced connection on submanifold

Definition

For a submanifold of a Riemannian manifold

Suppose $M$ is a Riemannian manifold, $N$ is a submanifold. Then $T N$ is a subbundle of the bundle $T M |_{N}$ . The Riemannian metric on $M$ naturally gives a notion of "projection" from $T M |_{N}$ to $T N$ .

Then, given any linear connection $\nabla$ on $M$ , we obtain an induced connection $\nabla |_{N}$ on $N$ . The induced connection is defined as follows. For $X, Y \in Γ (T M)$ , we have:

$(\nabla |_{N})_{X} Y : = (\nabla_{X} Y)^{t a n}$

The definition packs the following steps:

First, using the tubular neighborhood theorem, consider a tubular neighborhood $U \supset N$ in $M$
Extend $X, Y$ to vector fields on $U$ by Fill this in later
Evaluate $\nabla_{X} Y$ on this open set $U$ , to get a vector field on $U$
Restrict the vector field to $N$ . This gives a section of the bundle $T M |_{N}$
Project this from $T M |_{N}$ to $T N$

Crudely speaking, we use the connection on $M$ , and project the output vector field we get, onto $T N$ .