Induced connection on submanifold

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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 \nabla on M, we obtain an induced connection \nabla|_N on N. The induced connection is defined as follows. For X,Y \in \Gamma(TM), we have:

(\nabla|_N)_X Y := (\nabla_XY)^{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 \nabla_XY 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.