On the frame bundle adapted to a submanifold

Let M be a submanifold of a Riemannian manifold ( N , g ) . M induces a subbundle O ( M , N ) of adapted frames over M of the bundle of orthonormal frames O ( N ) . The Riemannian metric g induces a natural metric on O ( N ) . We study the geometry of a submanifold O ( M , N ) in O ( N ) . We characterize the horizontal distribution of O ( M , N ) and state its correspondence with the horizontal lift in O ( N ) 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 .