%0 Journal Article
%A Deshmukh, S.
%A Al-Shaikh, S. B.
%T Tangent Bundle of the Hypersurfaces in a Euclidean Space
%J Iranian Journal of Mathematical Sciences and Informatics
%V 11
%N 1
%U http://ijmsi.ir/article-1-430-en.html
%R 10.7508/ijmsi.2016.01.002
%D 2016
%K Tangent bundle, Hypersurface, Kaehler manifold, Almost contact structure, Killing vector field, CR-Submanifold, Second fundamental form, Wiengarten map.,
%X Let $M$ be an orientable hypersurface in the Euclidean space $R^{2n}$ with induced metric $g$ and $TM$ be its tangent bundle. It is known that the tangent bundle $TM$ has induced metric $overline{g}$ as submanifold of the Euclidean space $R^{4n}$ which is not a natural metric in the sense that the submersion $pi :(TM,overline{g})rightarrow (M,g)$ is not the Riemannian submersion. In this paper, we use the fact that $R^{4n}$ is the tangent bundle of the Euclidean space $R^{2n}$ to define a special complex structure $overline{J}$ on the tangent bundle $R^{4n}$ so that $% (R^{4n},overline{J}$,$leftlangle ,rightrangle )$ is a Kaehler manifold, where $leftlangle ,rightrangle $ is the Euclidean metric which is also the Sasaki metric of the tangent bundle $R^{4n}$. We study the structure induced on the tangent bundle $(TM,overline{g})$ of the hypersurface $M$, which is a submanifold of the Kaehler manifold $(R^{4n},overline{J}$,$% leftlangle ,rightrangle )$. We show that the tangent bundle $TM$ is a CR-submanifold of the Kaehler manifold $(R^{4n},overline{J}$,$leftlangle ,rightrangle )$. We find conditions under which certain special vector fields on the tangent bundle $(TM,overline{g})$ are Killing vector fields. It is also shown that the tangent bundle $TS^{2n-1}$ of the unit sphere $% S^{2n-1}$ admits a Riemannian metric $overline{g}$ and that there exists a nontrivial Killing vector field on the tangent bundle $(TS^{2n-1},% overline{g})$.
%> http://ijmsi.ir/article-1-430-en.pdf
%P 13-26
%& 13
%!
%9 Research paper
%L A-10-469-1
%+ King Saud University
%G eng
%@ 1735-4463
%[ 2016