Atlas: Some remarks on Kenmotsu manifolds by Tran Quoc Binh

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International Congress on Differential Geometry in memory of Alfred Gray (1939-1998)
September 18-23, 2000
Universidad del País Vasco
Bilbao, Spain

Organizers
M. Fernández (chairman), R. Ibáñez, M. Macho-Stadler

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Some remarks on Kenmotsu manifolds
by
Tran Quoc Binh
Debrecen University, Hungary

Poster

A (2n+1)-dimensional Riemannian manifold (M, g) is said to be a Kenmotsu manifold if it admits a vector field \xi, a 1-form \eta and an endomorphism j of its tangent bundle TM for which:

1)

(j, \eta, \xi, g) is metric almost contact structure, i.e.: j²=\textid +\eta\otimes\xi, \eta(\xi)=1, j\xi = 0, \eta o j = 0,

g(jX, jY)=g(X, Y)-\eta(X)\eta(Y), \eta(X)=g(X, \xi)

and

2) (Ñ_X j)Y = -g(X, jY)\xi-\eta(X) jX

Ñ_X\xi = X-\eta(X)\xi

for any X, Y vector fields on M, where Ñ denotes the Riemannian connection of g.

Kenmotsu manifold is a nice example of almost contact manifold, which is not a K-contact (and hence not a Sasakian-manifold), as shown by Kenmotsu [1].

In the present talk we show (among others) the following theorems:

Theorem 1 For a Kenmotsu manifold the following conditions are equivalent:

1) M is of constant curvature -1

2) M is locally symmetric, i.e.: ÑR=0

3) M is almost symmetric, i.e.: R(X, Y)R=0.

4) R(X, \xi)R=0 for any X.

Theorem 2 In a Kenmotsu manifold the following conditions are equivalent:

1) M is an Einstein space with S=-2ng

2) ÑS=0

3) R(X, Y)S=0 anf X, Y

4) R(X, \xi)S=0 for any X.

where S denotes the Ricci curvature tensor.

These are analogous to Tanno's theorems [2] for K-contact manifolds.

References [1] K. Kenmotsu, A class of almost contact Riemannian manifolds, Tohoku Math. Journ. 2 , 1972 , 93-103

[2] S. Tanno, Isometric immersions of Sasakian manifolds in spheres, Kodai Math. Sem. Rep. 21, 1969 , 448-458

Date received: June 1, 2000