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Conformal gradient vector fields on a compact Riemannian manifold

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Deshmukh S., Al-Solamy F.

Colloquium Mathematicum

2008

tom 112

nr 1

157-161

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 have positive scalar curvature and ultimately to be isometric to Sⁿ(c), where n(n-1)c is the scalar curvature of the manifold.

A note on conformal vector fields on a Riemannian manifold

100%

Deshmukh S., Al-Solamy F.

Colloquium Mathematicum

2014

tom 136

nr 1

65-73

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 using these vector fields.

Ograniczanie wyników

2 Colloquium Mathematicum

2 Al-Solamy F.

2 Deshmukh S.

1 2014

1 2008

Wyniki wyszukiwania

Conformal gradient vector fields on a compact Riemannian manifold

A note on conformal vector fields on a Riemannian manifold