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Conformal Vector Fields and the De-Rham Laplacian on a Riemannian Manifold

“…It is clear that such characterizations provide us a better insight of the relationship between differential equations and vector fields on Riemannian manifolds. In particular, it is worth mentioning that as immediate applications of the results, we obtain not only characterizations but also obstructions to the existence of certain nontrivial solutions to some (partial) differential equations on spaces of great interest in differential geometry, like Euclidean spheres, complex and quaternion projective spaces (see, e.g, [25]). Applications in physics are also notable, as many complicated physical problems are modeled through differential equations on certain (pseudo)-Riemannian manifolds (see, e.g, the recent books [32,33]).…”

“…where ∇ is the Riemannian connection and {e 1 , ..., e m } is a local orthonormal frame on M, m = dim M. Rough Laplace operator is used in finding characterizations of spheres as well as of Euclidean spaces (cf. [17,25]). Recall that the de-Rham Laplace operator : X(M) → X(M) on a Riemannian manifold (M, g) is defined by (cf.…”

Soliton-Type Equations on a Riemannian Manifold

“…hence, Hess( f ) 2 = nβ 2 and ∆( f ) = nβ, which implies the equality case in Schwartz's inequality. Note that in [8][9][10][11][12], the authors proved that a non-constant function f on a complete n-dimensional Riemannian manifold (M, g) satisfies Equation ( 8) for β a negative constant if and only if M is isometric to the n-dimensional Euclidean space. In [13], the authors proved that if Equation ( 8) holds with β a function, then (M, g) is locally a warped product (a, b) × h N n−1 .…”

“…Proposition 3. Let (M, g) be an n-dimensional compact Riemannian manifold and let f be a smooth function on M satisfying Equation (8). If Ric(∇ f , ∇ f ) ≤ 0, then ∇ f ∈ ker Q and Hess( f ) = 0.…”

On Differential Equations Characterizing Legendrian Submanifolds of Sasakian Space Forms

“…if and only if (N n , g) is isometric to the standard sphere S n . Such characterizations of complete spaces are of great interest and they were investigated by many geometers (see [3][4][5][6][7][8][9][10][11][12]). For example, Tashiro [13] has shown that the Euclidean spaces R n are characterized by a differential equation ∇ 2 ψ = cg, where c is a positive constant.…”