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Difference between revisions of "P-Sasakian manifold"

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A manifold similar to a [[Sasakian manifold|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|complex manifold]]).
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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|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
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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}
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\begin{equation}
 +
\label{eq1}
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\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,
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\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|Riemannian connection]] with respect to $g$, respectively, is called a P-Sasakian manifold [[#References|[a3]]].
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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 [[#References|[a3]]].
  
 
The structure group of the [[Tangent bundle|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 $n=\text{dim}M$.
 
The structure group of the [[Tangent bundle|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 $n=\text{dim}M$.
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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 $H^n$.
 
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 $f(x)=e^{-2x}$.
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The warped product $\mathbf{R}\times_f T^{n-1}$ of a real line $\mathbf{R}$ and an $(n-1)$-dimensional flat torus $T^{n-1}$, with $f(x)=e^{-2x}$.
  
 
==Properties.==
 
==Properties.==

Latest revision as of 15:54, 16 March 2024

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} \label{eq1} \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 $\mathbf{R}\times_f T^{n-1}$ of a real line $\mathbf{R}$ 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
How to Cite This Entry:
P-Sasakian manifold. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=P-Sasakian_manifold&oldid=55645
This article was adapted from an original article by I. Mihai (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article