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.