Iranian Journal of Mathematical Sciences and Informatics مجله علوم ریاضی و انفورماتیک ایرانیان IJMSI Basic Sciences http://ijmsi.ir 1 admin 1735-4463 2008-9473 8 7 14 8888 13 en jalali 1395 1 1 gregorian 2016 4 1 11 1 online 1 fulltext
en Tangent Bundle of the Hypersurfaces in a Euclidean Space تخصصي Special پژوهشي Research paper <p>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<br> 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}\$,\$%<br> leftlangle ,rightrangle )\$. We show that the tangent bundle \$TM\$ is a CR-submanifold of the Kaehler manifold&nbsp; \$(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})\$.</p> Tangent bundle, Hypersurface, Kaehler manifold, Almost contact structure, Killing vector field, CR-Submanifold, Second fundamental form, Wiengarten map. 13 26 http://ijmsi.ir/browse.php?a_code=A-10-469-1&slc_lang=en&sid=1 S. Deshmukh shariefd@ksu.edu.sa `10031947532846002915` 10031947532846002915 Yes King Saud University S. B. Al-Shaikh `10031947532846002916` 10031947532846002916 No King Saud University