$V$-minimal submanifolds
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866917775318974464 |
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| author | Aprodu, Monica Alice |
| author_facet | Aprodu, Monica Alice |
| contents | We introduce the notion of $V$-minimality, for $V$ a smooth vector field on a Riemannian manifold, a natural extension of the classical notion of minimality, and we prove several basic properties. One featured example is given for locally conformal Kaehler (l.c.K) manifolds. It is well-known that in general, complex submanifolds in non-Kaehler l.c.K manifolds are not minimal. We prove that, however, they are $V$-minimal for $V$ a suitable multiple of the Lee vector field. Extending some results from \cite{AAB}, to emphasis the utility of this notion, we prove that a PHH submersion is $V$-harmonic if and only if it has minimal fibres and a PHH $V$-harmonic submersion pulls back complex submanifolds to $V$ minimal submanifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_02104 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | $V$-minimal submanifolds Aprodu, Monica Alice Differential Geometry We introduce the notion of $V$-minimality, for $V$ a smooth vector field on a Riemannian manifold, a natural extension of the classical notion of minimality, and we prove several basic properties. One featured example is given for locally conformal Kaehler (l.c.K) manifolds. It is well-known that in general, complex submanifolds in non-Kaehler l.c.K manifolds are not minimal. We prove that, however, they are $V$-minimal for $V$ a suitable multiple of the Lee vector field. Extending some results from \cite{AAB}, to emphasis the utility of this notion, we prove that a PHH submersion is $V$-harmonic if and only if it has minimal fibres and a PHH $V$-harmonic submersion pulls back complex submanifolds to $V$ minimal submanifolds. |
| title | $V$-minimal submanifolds |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2306.02104 |