It is proved that if an n-dimensional compact connected Riemannian manifold (M,g) with Ricci curvature Ric satisfying
0 < Ric ≤ (n-1)(2-nc/λ₁)c
for a constant c admits a nonzero conformal gradient vector field, then it is isometric to Sⁿ(c), where λ₁ is the first nonzero eigenvalue of the Laplacian operator on M. Also, it is observed that existence of a nonzero conformal gradient vector field on an n-dimensional compact connected Einstein manifold forces it to...
We consider an n-dimensional compact Riemannian manifold (M,g) and show that the presence of a non-Killing conformal vector field ξ on M that is also an eigenvector of the Laplacian operator acting on smooth vector fields with eigenvalue λ > 0, together with an upper bound on the energy of the vector field ξ, implies that M is isometric to the n-sphere Sⁿ(λ). We also introduce the notion of φ-analytic conformal vector fields, study their properties, and obtain a characterization of n-spheres...
2000 Mathematics Subject Classification: 53C40, 53C25.
In the present note, it is proved that there donot exist warped product semi-slant submanifolds in a Sasakian manifold other than contact CR-warped product submanifolds and thus the results obtained in [8] are generalized.
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