The Geometry and the Laplace Operator on the Exterior 2-Forms on a Compact Riemannian Manifold

1979 ◽  
Vol 73 (1) ◽  
pp. 109 ◽  
Author(s):  
GR. Tsagas ◽  
C. Kockinos
Keyword(s):  
Riemannian Manifold ◽  
Laplace Operator ◽  
Compact Riemannian Manifold ◽  
The Laplace Operator

scholarly journals A Note on Killing Calculus on Riemannian Manifolds

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 307
Author(s):  
Sharief Deshmukh ◽  
Amira Ishan ◽  
Suha B. Al-Shaikh ◽  
Cihan Özgür
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Laplace Operator ◽  
Compact Riemannian Manifold ◽  
Killing Vector ◽  
Adjoint Operator ◽  
Dimensional Sphere ◽  
Killing Vector Field ◽  
Smooth Functions ◽  
The Laplace Operator

In this article, it has been observed that a unit Killing vector field ξ on an n-dimensional Riemannian manifold (M,g), influences its algebra of smooth functions C∞(M). For instance, if h is an eigenfunction of the Laplace operator Δ with eigenvalue λ, then ξ(h) is also eigenfunction with same eigenvalue. Additionally, it has been observed that the Hessian Hh(ξ,ξ) of a smooth function h∈C∞(M) defines a self adjoint operator ⊡ξ and has properties similar to most of properties of the Laplace operator on a compact Riemannian manifold (M,g). We study several properties of functions associated to the unit Killing vector field ξ. Finally, we find characterizations of the odd dimensional sphere using properties of the operator ⊡ξ and the nontrivial solution of Fischer–Marsden differential equation, respectively.

scholarly journals On Killing Vector Fields on Riemannian Manifolds

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 259
Author(s):  
Sharief Deshmukh ◽  
Olga Belova
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Laplace Operator ◽  
Ricci Curvature ◽  
Riemannian Manifolds ◽  
Smooth Function ◽  
Compact Riemannian Manifold ◽  
Killing Vector ◽  
Killing Vector Field ◽  
The Laplace Operator

We study the influence of a unit Killing vector field on geometry of Riemannian manifolds. For given a unit Killing vector field w on a connected Riemannian manifold (M,g) we show that for each non-constant smooth function f∈C∞(M) there exists a non-zero vector field wf associated with f. In particular, we show that for an eigenfunction f of the Laplace operator on an n-dimensional compact Riemannian manifold (M,g) with an appropriate lower bound on the integral of the Ricci curvature S(wf,wf) gives a characterization of the odd-dimensional unit sphere S2m+1. Also, we show on an n-dimensional compact Riemannian manifold (M,g) that if there exists a positive constant c and non-constant smooth function f that is eigenfunction of the Laplace operator with eigenvalue nc and the unit Killing vector field w satisfying ∇w2≤(n−1)c and Ricci curvature in the direction of the vector field ∇f−w is bounded below by n−1c is necessary and sufficient for (M,g) to be isometric to the sphere S2m+1(c). Finally, we show that the presence of a unit Killing vector field w on an n-dimensional Riemannian manifold (M,g) with sectional curvatures of plane sections containing w equal to 1 forces dimension n to be odd and that the Riemannian manifold (M,g) becomes a K-contact manifold. We also show that if in addition (M,g) is complete and the Ricci operator satisfies Codazzi-type equation, then (M,g) is an Einstein Sasakian manifold.

scholarly journals Some Inequalities for the Omori-Yau Maximum Principle

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Kyusik Hong
Keyword(s):  
Riemannian Manifold ◽  
Maximum Principle ◽  
Laplace Operator ◽  
Order Term ◽  
Complete Riemannian Manifold ◽  
Sufficient Condition ◽  
Zeroth Order ◽  
Exhaustion Function ◽  
Semielliptic Operator ◽  
The Laplace Operator

We generalize A. Borbély’s condition for the conclusion of the Omori-Yau maximum principle for the Laplace operator on a complete Riemannian manifold to a second-order linear semielliptic operatorLwith bounded coefficients and no zeroth order term. Also, we consider a new sufficient condition for the existence of a tamed exhaustion function. From these results, we may remark that the existence of a tamed exhaustion function is more general than the hypotheses in the version of the Omori-Yau maximum principle that was given by A. Ratto, M. Rigoli, and A. G. Setti.

Some Properties of the Eigenfunctions of The Laplace-Operator on Riemannian Manifolds

1949 ◽  
Vol 1 (3) ◽  
pp. 242-256 ◽  
Author(s):  
S. Minakshisundaram ◽  
Å. Pleijel
Keyword(s):  
Differential Equation ◽  
Riemannian Manifold ◽  
Laplace Operator ◽  
Riemannian Manifolds ◽  
Line Element ◽  
Metric Tensor ◽  
Local Coordinates ◽  
Image Position ◽  
The Laplace Operator ◽  
Beltrami Laplace Operator

Let V be a connected, compact, differentiable Riemannian manifold. If V is not closed we denote its boundary by S. In terms of local coordinates (xi), i = 1, 2, … Ν, the line-element dr is given by where gik (x1, x2, … xN) are the components of the metric tensor on V We denote by Δ the Beltrami-Laplace-Operator and we consider on V the differential equation (1) Δu + λu = 0.

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