On the frame bundle adapted to a submanifold

Let M be a submanifold of a Riemannian manifold . M induces a subbundle of adapted frames over M of the bundle of orthonormal frames . The Riemannian metric g induces a natural metric on . We study the geometry of a submanifold in . We characterize the horizontal distribution of and state its correspondence with the horizontal lift in induced by the Levi–Civita connection on N. In the case of extrinsic geometry, we show that minimality is equivalent to harmonicity of the Gauss map of the submanifold M with a deformed Riemannian metric. In the case of intrinsic geometry we compute the curvatures and compare this geometry with the geometry of M.