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Four‐dimensional contact C R ‐submanifolds in S 7 ( 1 )
Author(s) -
Djorić Mirjana,
Munteanu Marian Ioan,
Vrancken Luc
Publication year - 2017
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.201600437
Subject(s) - mathematics , submanifold , pure mathematics , tangent bundle , endomorphism , tangent , invariant (physics) , dimension (graph theory) , mathematical analysis , tangent space , geometry , mathematical physics
The analogue of C R ‐submanifolds in (almost) Kählerian manifolds is the concept of contact C R ‐submanifolds in Sasakian manifolds. These are submanifolds for which the structure vector field ξ is tangent to the submanifold and for which the tangent bundle of M can be decomposed as T ( M ) = H ( M ) ⊕ E ( M ) ⊕ R ξ , where H ( M ) is invariant with respect to the endomorphism φ and E ( M ) is antiinvariant with respect to φ. The lowest possible dimension for M in which this decomposition is non trivial is the dimension 4. In this paper we obtain a complete classification of four‐dimensional contact C R ‐submanifolds inS 5 ( 1 )andS 7 ( 1 )for which the second fundamental form restricted to H ( M ) and E ( M ) vanishes identically.