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by Sharief Deshmukh, Amira Ishan, Suha B. Al-Shaikh and Cihan Özgür

Mathematics 2021, 9(4), 307; https://doi.org/10.3390/math9040307 - 4 Feb 2021

Cited by 2 | Viewed by 2472

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^{\infty} (M)

. For instance, if h is [...] Read more.

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^{\infty} (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

H^{h} (ξ, ξ)

of a smooth function

h \in C^{\infty} (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. Full article

(This article belongs to the Special Issue Differential Geometry: Theory and Applications)

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