A remarkable property of concircular vector fields on a Riemannian manifold
Abstract
In this paper, we show that given a nontrivial concircular vector field $\boldsymbol{u}$ on a Riemannian manifold $(M,g)$ with potential function $f$, there exists a unique smooth function $\rho $ on $M$ that connects $\boldsymbol{u}$ to the gradient of potential function $\nabla f$, which we call the connecting function of the concircular vector field $\boldsymbol{u}$. Then this connecting function is shown to be a main ingredient in obtaining characterizations of $n$-sphere $\mathbf{S}^{n}(c)$ and the Euclidean space $\mathbf{E}^{n}$. We also show that the connecting function influences topology of the Riemannian manifold.
- Publication:
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arXiv e-prints
- Pub Date:
- November 2019
- DOI:
- 10.48550/arXiv.1912.01403
- arXiv:
- arXiv:1912.01403
- Bibcode:
- 2019arXiv191201403A
- Keywords:
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- Mathematics - Differential Geometry;
- 53C20 Primary;
- F.2.2
- E-Print:
- 10 pages