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 $(M,g)$ endowed with an endomorphism $\phi$ of the tangent bundle $TM$, a vector field $\xi$ and a $1$-form $\eta$ which satisfy the conditions

\begin{equation}\phi^2=I-\eta\bigotimes\xi,\\\eta(\xi)=1,\phi\xi=0,\eta\circ\phi=0,d\eta=0,g(\phi X,\phi Y)=h(X,Y)-\eta(X)\eta(Y),\\(\nabla_X\phi)Y=\{-g(X,Y)+2\eta(X)\eta(Y)\}\xi-\eta(Y)X,\end{equation}

for any vector fields $X$, $Y$ tangent to $M$, where $I$ and $\nabla$ denote the identity transformation on $TM$ and the Riemannian connection with respect to $g$, respectively, is called a P-Sasakian manifold [a3].

The structure group of the tangent bundle $TM$ is reducible to $O(h)\times O(n-h-1)\times1$, where $h$ is the multiplicity of the eigenvalue $1$ of the characteristic equation of $\phi$ and $n=\text{dim}M$.

Examples.

The hyperbolic $n$-space form $H^n$. As a model, one can take the upper half-space $x^n>0$ in the sense of Poincaré's representation (cf. also Poincaré model). The metric of $H^n$ is given by

\begin{equation}g_{ij}(x)=(x^n)^{-2}\delta_{ij}\end{equation}

where $x\in H^n$, $i,j=1,...,n$. The characteristic vector field $\xi=(0,...,0,x^n)$, and $\phi X=\nabla_X\xi$ for any vector field $X$ tangent to $H^n$.

The warped product $\textbf{R}\times_f T^{n-1}$ of a real line $\textbf{R}$ and an $(n-1)$-dimensional flat torus $T^{n-1}$, with $f(x)=e^{-2x}$.

Properties.

If a P-Sasakian manifold is a space form (cf. Space forms), then its sectional curvature is $-1$ [a2].

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

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

A projectively flat P-Sasakian manifold is a hyperbolic space form of constant curvature $-1$ [a5].

For the de Rham cohomology of a P-Sasakian manifold, the following result is known [a1]: Let $M$ be a compact P-Sasakian manifold such that the distribution annihilated by $\eta$ is minimal. Then the first Betti number $b_1$ 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