Induced connection on submanifold

Definition

For a submanifold of a Riemannian manifold

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

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

$(\mathrm{\nabla }{|}_N{)}_{X}Y:=({\mathrm{\nabla }}_{X}Y{)}^{tan}$

The definition has two key parts:

1. We use a connection on $TM$ to obtain a connection on $TM{|}_N$. 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 $U\supset N$ in $M$. Extend $X,Y$ to vector fields on $U$ by Fill this in later. Evaluate ${\mathrm{\nabla }}_{X}Y$ 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 $TM{|}_N$
2. We project this from $TM{|}_N$ to $TN$, using the Riemannian metric

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