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. |