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| On the Tangent Bundle of a Hypersurface in a Riemannian Manifold |
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Citation： |
Zhonghua HOU,Lei SUN.On the Tangent Bundle of a Hypersurface in a Riemannian Manifold[J].Chinese Annals of Mathematics B,2015,36(4):579~602 |
| Page view： 1180
Net amount： 885 |
Authors： |
Zhonghua HOU; Lei SUN; |
Foundation： |
supported by the National Natural Science Foundation of China (No. 61473059) and the
Fundamental Research Funds for the Central University (No.DUT11LK47). |
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| Abstract： |
Let $(M^n,g)$ and $(N^{n+1},G)$ be Riemannian manifolds. Let $TM^n$
and $TN^{n+1}$ be the associated tangent bundles. Let $f: (M^n,g)
\to (N^{n+1},G)$ be an isometrical immersion with $g=f^\ast G$,
$F=(f, df): (TM^n,\ov {g}) \to (TN^{n+1},G_s)$ be the isometrical
immersion with $\ov {g}=F^\ast G_s$ where $(df)_x: T_xM\to
T_{f(x)}N$ for any $x\in M$ is the differential map, and $G_s$ be
the Sasaki metric on $TN$ induced from $G$. This paper deals with
the geometry of $TM^n$ as a submanifold of $TN^{n+1}$ by the moving
frame method. The authors firstly study the extrinsic geometry of
$TM^n$ in $TN^{n+1}$. Then the integrability of the induced almost
complex structure of $TM$ is discussed. |
Keywords： |
Hypersurfaces, Tangent bundle, Mean curvature vector, Sasaki metric,
Almost complex structure, K¨ahlerian form |
Classification： |
32Q60, 53C42 |
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