2013/11/24 by Kamil Niedzialomski, Niedzialomski, Kamil
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #math.DG
paper · pdf · doi:10.48550/arxiv.1311.6172
19 pages, several improvements, minor corrections
arxiv created 2014/01/02 · arxiv updated 2014/01/03
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.