A Note on Killing Calculus on Riemannian Manifolds

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

doi:10.3390/math9040307

Sharief Deshmukh, Cihan Özgür + Show 2 more

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https://doi.org/10.3390/math9040307

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Abstract

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.

Highlights

A smooth vector field ξ on an n-dimensional Riemannian manifold ( M, g) is said to be a Killing vector field if its flow consists of isometries of ( M, g)
It is known that a nontrivial Killing vector field on a compact Riemannian manifold restricts its topology and geometry, for example, it does not allow the Riemannian manifold ( M, g) to have negative Ricci curvature and that if ( M, g) has positive sectional curvatures, its fundamental group contains a cyclic subgroup with constant index, depending only on the dimension of M
We use properties of the nontrivial solution h of the Fischer–Marsden equation on a compact Riemannian manifold ( M, g) with Killing vector field ξ and a suitable lower bound on the Ricci curvature Ric( gradh, gradh) to find a characterization of the unit sphere S2n+1 (c)