A new class of contact Riemannian manifolds
by
Jong Taek Cho
Chonnam National University, Korea

Poster

A Riemannian manifold M=(M, g) with Riemannian metric tenor g is called a locally symmetric space if M satisfies ÑR=0, where Ñ is the Levi-Civita connection. It was proved that a Sasakian manifold (or normal contact Riemannian manifold) which is locally symmetric must have constant curvature +1. This fact means that local symmetry is very strong condition for a Sasakian manifold. For this reason, T. Takahashi introduced the notion of Sasakian locally \phi-symmetric spaces which may be considered as the analogues of locally Hermitian symmetric spaces. A contact Riemannian locally \phi-symmetric space is defined as a generalization of the notion of the Sasakian locally \phi-symmetric spaces and investigated.

On the other hand, N.Tanaka defined the canonical affine connection on a nondegenerate integrable CR-manifold. S.Tanno defined the generalized Tanaka connection [^(Ñ)] on a contact Riemannian manifold and further, he proved that for a given contact Riemannian manifold M the associated CR-structure is strongly pseudo-convex integrable if and only if M satisfies the integrability condition, in which case the connection [^(Ñ)] coincides with Tanaka connection. Here we note that the normallity of a contact Riemannian structure implies the integrability of the associated CR-structure, but the converse does not always hold. The associated CR-structures of 3-dimensional contact Riemannian manifolds are always integrable. Also, we see that their associated CR-structures are integrable for contact Riemannian manifolds with the characteristic vector field \xi belonging to the (k, \mu)-nullity distribution.

Now we introduce a new class of contact Riemannian manifolds satisfying

( ^ Ñ   [(\gamma)\dot]  R)(·, × \gamma   ) × \gamma   =0 \tag C

for any unit [^(Ñ)]-geodesic \gamma ([^(Ñ)]_{[(\gamma)\dot]}[(\gamma)\dot]=0), where [^(Ñ)] is the generalized Tanaka connection. We observe that the geodesics of the Levi-Civita connection and the generalized Tanaka connection do not coincide in general, and further that a contact Riemannian manifold satisfies condition (C) for any [^(Ñ)]-geodesic \gamma if and only if the Jacobi operator field R_{[(\gamma)\dot]} is diagonalizable by a [^(Ñ)]-parallel orthonormal frame field along \gamma and its eigenvalues are constant along \gamma for any [^(Ñ)]-geodesic \gamma in the manifold. Furthermore, we can show that contact Riemannian manifolds with \xi belonging to the (k, 2)-nullity distribution (k =/= 1) including the standard contact Riemannian structure of the unit tangent sphere bundle T₁ M of M with constant curvature -1 are examples that are neither Sasakian nor locally symmetric but satisfy the condition (C) for any [^(Ñ)]-geodesic \gamma. Also, it is remarkable that a contact Riemannian manifold M of dimension >= 5 whose characteristic vector field \xi belonging to the (0, 0)-nullity distribution is locally symmetric, but M fails to satisfy the condition (C) for any [^(Ñ)]-geodesic \gamma. We have proved Theorem A. The standard contact Riemannian structure of the unit tangent sphere bundle T₁ M satisfies the condition (C) for any [^(Ñ)]-geodesic \gamma if and only if

(1) the base manifold M is of constant curvature 1 and T₁ M is Sasakian locally \phi-symmetric;

(2) the base manifold M is 2-dimensional and T₁ M is flat;

(3) the base manifold M is of constant curvature -1.

Moreover, we have given a local classification and a global classification of a 3-dimensional contact Riemannian manifold satisfying the condition (C) for any [^(Ñ)]-geodesic \gamma. More precisely, we have proved that Theorem B(local classification). Let M be a 3-dimensional complete contact Riemannian manifold. Then M satisfies the condition (C) for any [^(Ñ)]-geodesic \gamma if and only if M is locally isometric to one of the followings : (1) a Sasakian \phi-symmetric space; (2) SU(2) (or SO(3)), SL(2, R) (or O(1, 2)) with a left invariant contact metric which is not Sasakian, respectively; (3) a flat manifold. Theorem C(global classification). Let M be a complete and simply connected 3-dimensional contact Riemannian manifold. Then M satisfies the condition (C) for any [^(Ñ)]-geodesic \gamma if and only if M is isometric to one of the followings : (1) the standard unit sphere S³; SU(2), [(SL(2, R))\tilde] (the universal covering of SL(2, R)) or the Heisenberg group H with a left invariant Sasakian metric, respectively; (2) SU(2), [(SL(2, R))\tilde] with a left invariant contact metric which is not Sasakian, respectively; (3) R ³.

Date received: May 22, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadq-50.