A Short Note on a Mus-Cheeger-Gromoll Type Metric
In this paper, we first show that the complete lift $U^{c}$ to $TM$ of a vector field $U$ on $M$ is an infinitesimal fiber-preserving conformal transformation if and only if $U$ is an infinitesimal homothetic transformation of $(M,g)$. Here, $(M, g)$ is a Riemannian manifold and $TM$ is its tangent...
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| Format: | Article |
| Language: | English |
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Naim Çağman
2023-03-01
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| Series: | Journal of New Theory |
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| Online Access: | https://dergipark.org.tr/en/download/article-file/2616089 |
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| author | Murat Altunbaş |
| author_facet | Murat Altunbaş |
| author_sort | Murat Altunbaş |
| collection | DOAJ |
| description | In this paper, we first show that the complete lift $U^{c}$ to $TM$ of a vector field $U$ on $M$ is an infinitesimal fiber-preserving conformal transformation if and only if $U$ is an infinitesimal homothetic transformation of $(M,g)$. Here, $(M, g)$ is a Riemannian manifold and $TM$ is its tangent bundle with a Mus-Cheeger-Gromoll type metric $\tilde{g}$. Secondly, we search for some conditions under which $\left(\overset{h}{\nabla},\tilde{g}\right)$ is a Codazzi pair on $TM$ when $(\nabla, g)$ is a Codazzi pair on $M$ where $\overset{h}{\nabla}$ is the horizontal lift of a linear connection $\nabla$ on $M$. We finally discuss the need for further research. |
| format | Article |
| id | doaj-art-f2dd28b96bbf4b58996dbff5a8ea6479 |
| institution | DOAJ |
| issn | 2149-1402 |
| language | English |
| publishDate | 2023-03-01 |
| publisher | Naim Çağman |
| record_format | Article |
| series | Journal of New Theory |
| spelling | doaj-art-f2dd28b96bbf4b58996dbff5a8ea64792025-08-20T02:45:06ZengNaim ÇağmanJournal of New Theory2149-14022023-03-01421710.53570/jnt.11670102425A Short Note on a Mus-Cheeger-Gromoll Type MetricMurat Altunbaş0https://orcid.org/0000-0002-0371-9913ERZINCAN UNIVERSITYIn this paper, we first show that the complete lift $U^{c}$ to $TM$ of a vector field $U$ on $M$ is an infinitesimal fiber-preserving conformal transformation if and only if $U$ is an infinitesimal homothetic transformation of $(M,g)$. Here, $(M, g)$ is a Riemannian manifold and $TM$ is its tangent bundle with a Mus-Cheeger-Gromoll type metric $\tilde{g}$. Secondly, we search for some conditions under which $\left(\overset{h}{\nabla},\tilde{g}\right)$ is a Codazzi pair on $TM$ when $(\nabla, g)$ is a Codazzi pair on $M$ where $\overset{h}{\nabla}$ is the horizontal lift of a linear connection $\nabla$ on $M$. We finally discuss the need for further research.https://dergipark.org.tr/en/download/article-file/2616089codazzi pairinfinitesimal fiber-preserving conformal transformationinfinitesimal homothetic transformationmus-cheeger-gromoll type metrictangent bundle |
| spellingShingle | Murat Altunbaş A Short Note on a Mus-Cheeger-Gromoll Type Metric Journal of New Theory codazzi pair infinitesimal fiber-preserving conformal transformation infinitesimal homothetic transformation mus-cheeger-gromoll type metric tangent bundle |
| title | A Short Note on a Mus-Cheeger-Gromoll Type Metric |
| title_full | A Short Note on a Mus-Cheeger-Gromoll Type Metric |
| title_fullStr | A Short Note on a Mus-Cheeger-Gromoll Type Metric |
| title_full_unstemmed | A Short Note on a Mus-Cheeger-Gromoll Type Metric |
| title_short | A Short Note on a Mus-Cheeger-Gromoll Type Metric |
| title_sort | short note on a mus cheeger gromoll type metric |
| topic | codazzi pair infinitesimal fiber-preserving conformal transformation infinitesimal homothetic transformation mus-cheeger-gromoll type metric tangent bundle |
| url | https://dergipark.org.tr/en/download/article-file/2616089 |
| work_keys_str_mv | AT murataltunbas ashortnoteonamuscheegergromolltypemetric AT murataltunbas shortnoteonamuscheegergromolltypemetric |