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Almost-contact manifold
In the mathematical field of
differential geometry , a Kenmotsu manifold is an
almost-contact manifold endowed with a certain kind of
Riemannian metric . They are named after the Japanese mathematician Katsuei Kenmotsu.
Definitions
Let
(
M
,
φ
,
ξ
,
η
)
{\displaystyle (M,\varphi ,\xi ,\eta )}
be an
almost-contact manifold . One says that a
Riemannian metric
g
{\displaystyle g}
on
M
{\displaystyle M}
is adapted to the almost-contact structure
(
φ
,
ξ
,
η
)
{\displaystyle (\varphi ,\xi ,\eta )}
if:
g
i
j
ξ
j
=
η
i
g
p
q
φ
i
p
φ
j
q
=
g
i
j
−
η
i
η
j
.
{\displaystyle {\begin{aligned}g_{ij}\xi ^{j}&=\eta _{i}\\g_{pq}\varphi _{i}^{p}\varphi _{j}^{q}&=g_{ij}-\eta _{i}\eta _{j}.\end{aligned}}}
That is to say that, relative to
g
p
,
{\displaystyle g_{p},}
the vector
ξ
p
{\displaystyle \xi _{p}}
has length one and is orthogonal to
ker
(
η
p
)
;
{\displaystyle \ker \left(\eta _{p}\right);}
furthermore the restriction of
g
p
{\displaystyle g_{p}}
to
ker
(
η
p
)
{\displaystyle \ker \left(\eta _{p}\right)}
is a
Hermitian metric relative to the
almost-complex structure
φ
p
|
ker
(
η
p
)
.
{\displaystyle \varphi _{p}{\big \vert }_{\ker \left(\eta _{p}\right)}.}
One says that
(
M
,
φ
,
ξ
,
η
,
g
)
{\displaystyle (M,\varphi ,\xi ,\eta ,g)}
is an
almost-contact metric manifold .
An almost-contact metric manifold
(
M
,
φ
,
ξ
,
η
,
g
)
{\displaystyle (M,\varphi ,\xi ,\eta ,g)}
is said to be a Kenmotsu manifold if
∇
i
φ
j
k
=
−
η
j
φ
i
k
−
g
i
p
φ
j
p
ξ
k
.
{\displaystyle \nabla _{i}\varphi _{j}^{k}=-\eta _{j}\varphi _{i}^{k}-g_{ip}\varphi _{j}^{p}\xi ^{k}.}
References
Sources
Basic concepts Types of manifolds Main results Generalizations Applications
Basic concepts Main results
(list)
Maps Types of manifolds
Tensors
Related Generalizations