On the Sasaki metric of the normal bundle of a submanifold in Riemannian space. (Russian) Zbl 0633.53071

Borisenko, A. A.; Yampol’skij, A. L.

Mat. Sb., N. Ser. 134(176), No. 2(10), 158-176 (1987).

If F is a submanifold of a Riemannian manifold M, then the normal bundle \(\pi\) : NF\(\to F\) carries the connection D induced by the Levi-Civita connection on M. D induces the decomposition \(T(NF)=\pi^*TF\oplus \pi^*NF\) which allows to construct - in a canonical way - a Riemannian metric g on NF. g is called the Sasaki metric of NF. The curvature tensor of g is studied here. The authors give necessary and (or) sufficient conditions for the flatness of g and sign conditions for the sectional curvature of (NF,g). The proofs are based on the calculation analogous to that of O. Kowalski [J. Reine Angew. Math. 250, 124-129 (1971; Zbl 0222.53044)].

Reviewer: P.Walczak

MSC:

53C25	Special Riemannian manifolds (Einstein, Sasakian, etc.)
53B25	Local submanifolds

Keywords:

submanifold; normal bundle; Sasaki metric; sectional curvature

Citations:

Zbl 0222.53044