# How to induce a connection on a submanifold?

Suppose an affine connection is given on a smooth manifold $M$ and let $N\subset M$ be an embedded submanifold. Is there a canonical way of defining an induced connection on $N$?

In classical differential geometry of smooth surfaces in Euclidean 3-space, the corresponding construction is that of covariant derivative (cfr. Do Carmo, Differential geometry of curves and surfaces §4-4). The covariant derivative of a vector field along a curve on the surface is defined as the orthogonal projection of the ordinary Euclidean derivative onto the plane tangent to the surfaces.

I wonder how (and if) this can be ported to the language of connections.Wikipedia's entry does something like that by means of the Riemannian structure: I wonder if this extra structure is really necessary.

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A connection more or less tells you how to move tangent vectors consistently from $T_p M$ to $T_q M$. The point about the Riemannian metric is, that it defines orthogonal projections $T_p M \rightarrow T_p N$ if $p\in N$, providing a natural means to induce a connection on $N$. Unless you have some natural projection you will have difficulties to find a canonical way to define a connection on $N$. – user20266 Jun 7 '12 at 15:12

Instead, you can consider the following for codimension 1 submanifolds: given $\tau:N\to M$ an embedding and let $v$ be a vector field on $M$ along $N$ that is transverse to $N$, then $(\nabla,v)$ on $M$ together induces a connection on $N$. For $(X,Y)$ vector fields on $N$, we can define $$D^{(v)}_X Y = [\nabla_{\tau_*X}\tau_*Y]$$ where $[W]$ for $W\in T_pM$, $p\in \tau(N)$ is defined by $\tau_*[W] - W = \lambda v$ for some $\lambda\in\mathbb{R}$. For higher codimension case you need more (linearly independent) vector fields. In the Riemannian case, $v$ is canonically chosen to be the unit normal vector to $N$ (or in higher codimension, a family that spans the normal bundle).