On the frame bundle adapted to a submanifold

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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). Riemannian metric g induces 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 deformed Riemannian metric. In the case of intrinsic geometry we compute the curvatures.