Conformal gradient vector fields on a compact Riemannian manifold

Sharief Deshmukh, and Falleh Al-Solamy. "Conformal gradient vector fields on a compact Riemannian manifold." Colloquium Mathematicae 112.1 (2008): 157-161. <http://eudml.org/doc/283671>.

@article{ShariefDeshmukh2008,
abstract = { 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. },
author = {Sharief Deshmukh, Falleh Al-Solamy},
journal = {Colloquium Mathematicae},
keywords = {Ricci curvature; conformal gradient vector field; eigenvalue of the Laplacian operator},
language = {eng},
number = {1},
pages = {157-161},
title = {Conformal gradient vector fields on a compact Riemannian manifold},
url = {http://eudml.org/doc/283671},
volume = {112},
year = {2008},
}

TY - JOUR
AU - Sharief Deshmukh
AU - Falleh Al-Solamy
TI - Conformal gradient vector fields on a compact Riemannian manifold
JO - Colloquium Mathematicae
PY - 2008
VL - 112
IS - 1
SP - 157
EP - 161
AB - 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.
LA - eng
KW - Ricci curvature; conformal gradient vector field; eigenvalue of the Laplacian operator
UR - http://eudml.org/doc/283671
ER -