P-Sasakian manifold

A manifold similar to a Sasakian manifold. However, by its topological and geometric properties, such a manifold is closely related to a product manifold (unlike the Sasakian manifold, which is related to a complex manifold).

A Riemannian manifold endowed with an endomorphism of the tangent bundle , a vector field and a -form which satisfy the conditions

for any vector fields , tangent to , where and denote the identity transformation on and the Riemannian connection with respect to , respectively, is called a P-Sasakian manifold [a3].

The structure group of the tangent bundle is reducible to , where is the multiplicity of the eigenvalue of the characteristic equation of and .

Examples.

The hyperbolic -space form . As a model, one can take the upper half-space in the sense of Poincaré's representation (cf. also Poincaré model). The metric of is given by

where , . The characteristic vector field , and for any vector field tangent to .

The warped product of a real line and an -dimensional flat torus , with .

Properties.

If a P-Sasakian manifold is a space form (cf. Space forms), then its sectional curvature is [a2].

The characteristic vector field of a P-Sasakian manifold is an exterior concurrent vector field [a2].

On a compact orientable P-Sasakian manifold, the characteristic vector field is harmonic [a5].

A projectively flat P-Sasakian manifold is a hyperbolic space form of constant curvature [a5].

For the de Rham cohomology of a P-Sasakian manifold, the following result is known [a1]: Let be a compact P-Sasakian manifold such that the distribution annihilated by is minimal. Then the first Betti number does not vanish.

References

[a1]	I. Mihai, R. Rosca, L. Verstraelen, "Some aspects of the differential geometry of vector fields" , PADGE , 2 , KU Leuven&KU Brussel (1996)
[a2]	R. Rosca, "On para Sasakian manifolds" Rend. Sem. Mat. Messina , 1 (1991) pp. 201–216
[a3]	I. Sato, "On a structure similar to the almost contact structure I; II" Tensor N.S. , 30/31 (1976/77) pp. 219–224; 199–205
[a4]	I. Sato, "On a Riemannian manifold admitting a certain vector field" Kodai Math. Sem. Rep. , 29 (1978) pp. 250–260
[a5]	I. Sato, K. Matsumoto, "On P-Sasakian manifolds satisfying certain conditions" Tensor N.S. , 33 (1979) pp. 173–178