@article{article_987379, title={Notes on the second-order tangent bundles with the deformed Sasaki metric}, journal={Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics}, volume={71}, pages={502–517}, year={2022}, DOI={10.31801/cfsuasmas.987379}, author={Karaca, Kübra and Gezer, Aydın and Mağden, Abdullah}, keywords={Second-order tangent bundle, deformed Sasaki metric, weakly symmetric, curvature tensor field, lifts}, abstract={The paper deals with the second-order tangent bundle  <span class="MathJax_Preview" style="color:inherit;"> </span> <span class="mjx-chtml MathJax_CHTML" style="font-size:123%;"> <span class="mjx-math"> <span class="mjx-mrow"> <span class="mjx-msubsup"> <span class="mjx-base" style="margin-right:-.12em;"> <span class="mjx-mi"> <span class="mjx-char MJXc-TeX-math-I" style="padding-top:.441em;padding-bottom:.253em;padding-right:.12em;">T </span> </span> </span> <span class="mjx-sup" style="font-size:70.7%;vertical-align:.513em;padding-left:.291em;padding-right:.071em;"> <span class="mjx-texatom"> <span class="mjx-mrow"> <span class="mjx-mn"> <span class="mjx-char MJXc-TeX-main-R" style="padding-top:.378em;padding-bottom:.316em;">2 </span> </span> </span> </span> </span> </span> <span class="mjx-mi"> <span class="mjx-char MJXc-TeX-math-I" style="padding-top:.441em;padding-bottom:.253em;padding-right:.081em;">M </span> </span> </span> </span> <span class="MJX_Assistive_MathML">T2M </span> </span> with the deformed Sasaki metric <span class="MathJax_Preview" style="color:inherit;"> </span> <span class="mjx-chtml MathJax_CHTML" style="font-size:123%;"> <span class="mjx-math"> <span class="mjx-mrow"> <span class="mjx-texatom"> <span class="mjx-mrow"> <span class="mjx-munderover"> <span class="mjx-stack"> <span class="mjx-over" style="height:.096em;padding-bottom:.06em;padding-left:.045em;"> <span class="mjx-mo" style="vertical-align:top;"> <span class="mjx-char MJXc-TeX-main-R" style="padding-top:.316em;padding-bottom:.316em;">¯ </span> </span> </span> <span class="mjx-op"> <span class="mjx-mi"> <span class="mjx-char MJXc-TeX-math-I" style="padding-top:.191em;padding-bottom:.503em;padding-right:.003em;">g </span> </span> </span> </span> </span> </span> </span> </span> </span> <span class="MJX_Assistive_MathML">g¯ </span> </span> over an <i>n </i>−dimensional Riemannian manifold  <span class="mjx-chtml MathJax_CHTML" style="font-size:123%;"> <span class="MJX_Assistive_MathML">g <span style="font-size:13px;">over an </span> <i style="font-size:13px;">n </i> <span style="font-size:13px;">−dimensional Riemannian manifold  </span> </span> </span> <span class="mjx-chtml MathJax_CHTML" style="font-size:123%;"> <span class="mjx-math"> <span class="mjx-mrow"> <span class="mjx-mo"> <span class="mjx-char MJXc-TeX-main-R" style="padding-top:.441em;padding-bottom:.566em;">( </span> </span> <span class="mjx-mi"> <span class="mjx-char MJXc-TeX-math-I" style="padding-top:.441em;padding-bottom:.253em;padding-right:.081em;">M </span> </span> <span class="mjx-mo"> <span class="mjx-char MJXc-TeX-main-R" style="margin-top:-.184em;padding-bottom:.566em;">, </span> </span> <span class="mjx-mi MJXc-space1"> <span class="mjx-char MJXc-TeX-math-I" style="padding-top:.191em;padding-bottom:.503em;padding-right:.003em;">g </span> </span> <span class="mjx-mo"> <span class="mjx-char MJXc-TeX-main-R" style="padding-top:.441em;padding-bottom:.566em;">) </span> </span> </span> </span> <span class="MJX_Assistive_MathML">(M,g) </span> </span>. We calculate all Riemannian curvature tensor fields of the deformed Sasaki metric <span class="MathJax_Preview" style="color:inherit;"> </span> <span class="mjx-chtml MathJax_CHTML" style="font-size:123%;"> <span class="mjx-math"> <span class="mjx-mrow"> <span class="mjx-texatom"> <span class="mjx-mrow"> <span class="mjx-munderover"> <span class="mjx-stack"> <span class="mjx-over" style="height:.096em;padding-bottom:.06em;padding-left:.045em;"> <span class="mjx-mo" style="vertical-align:top;"> <span class="mjx-char MJXc-TeX-main-R" style="padding-top:.316em;padding-bottom:.316em;">¯ </span> </span> </span> <span class="mjx-op"> <span class="mjx-mi"> <span class="mjx-char MJXc-TeX-math-I" style="padding-top:.191em;padding-bottom:.503em;padding-right:.003em;">g </span> </span> </span> </span> </span> </span> </span> </span> </span> <span class="MJX_Assistive_MathML">g¯ </span> </span> and search Einstein property of  <span class="mjx-chtml MathJax_CHTML" style="font-size:123%;"> <span class="MJX_Assistive_MathML"> <span style="font-size:13px;">and search Einstein property of  </span> </span> </span> <span class="mjx-chtml MathJax_CHTML" style="font-size:123%;"> <span class="mjx-math"> <span class="mjx-mrow"> <span class="mjx-msubsup"> <span class="mjx-base" style="margin-right:-.12em;"> <span class="mjx-mi"> <span class="mjx-char MJXc-TeX-math-I" style="padding-top:.441em;padding-bottom:.253em;padding-right:.12em;">T </span> </span> </span> <span class="mjx-sup" style="font-size:70.7%;vertical-align:.513em;padding-left:.291em;padding-right:.071em;"> <span class="mjx-texatom"> <span class="mjx-mrow"> <span class="mjx-mn"> <span class="mjx-char MJXc-TeX-main-R" style="padding-top:.378em;padding-bottom:.316em;">2 </span> </span> </span> </span> </span> </span> <span class="mjx-mi"> <span class="mjx-char MJXc-TeX-math-I" style="padding-top:.441em;padding-bottom:.253em;padding-right:.081em;">M </span> </span> </span> </span> <span class="MJX_Assistive_MathML">T2M </span> </span>. Also the weakly symmetry properties of the deformed Sasaki metric are presented.}, number={2}, publisher={Ankara University}