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On the frame bundle adapted to a submanifold
Author(s) -
Niedziałomski Kamil
Publication year - 2015
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201400041
Subject(s) - submanifold , mathematics , normal bundle , fundamental theorem of riemannian geometry , connection (principal bundle) , geometry , metric connection , parallel transport , mathematical analysis , pure mathematics , frame bundle , levi civita connection , bundle , riemannian geometry , metric (unit) , topology (electrical circuits) , ricci curvature , combinatorics , vector bundle , curvature , materials science , composite material , operations management , economics
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 .