Abstract

In the present paper, we study a Riemannian submersion $\pi$ from a Riemann soliton $(M_1,g,\xi,\lambda)$ onto a Riemannian manifold $(M_2,g^{'})$. We first calculate the sectional curvatures of any fibre of $\pi$ and the base manifold $M_2$. Using them, we give some necessary and sufficient conditions for which the Riemann soliton $(M_1,g,\xi,\lambda)$ is shrinking, steady or expanding. Also, we deal with the potential field $\xi$ of such a Riemann soliton is conformal and obtain some characterizations about the extrinsic vertical and horizontal sectional curvatures of $\pi$.

Keywords: Riemannian submersion; Riemann soliton; sectional curvature.

MSC: 53C40, 32Q15

DOI: 10.57016/MV-o1hsah21

Pages:  257--265     

Volume  76 ,  Issue  4 ,  2024