Hodge Decomposition of Conformal Vector Fields on a Riemannian Manifold and Its Applications

Hanan Alohali, Sharief Deshmukh, Bang-Yen Chen () and Hemangi Madhusudan Shah
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Hanan Alohali: Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
Sharief Deshmukh: Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
Bang-Yen Chen: Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
Hemangi Madhusudan Shah: Homi Bhabha National Institute, Harish-Chandra Research Institute, Jhunsi, Allahabad 211019, Uttar Pradesh, India

Mathematics, 2024, vol. 12, issue 17, 1-14

Abstract: For a compact Riemannian m -manifold ( M m , g ) , m > 1 , endowed with a nontrivial conformal vector field ζ with a conformal factor σ , there is an associated skew-symmetric tensor φ called the associated tensor, and also, ζ admits the Hodge decomposition ζ = ζ ¯ + ∇ ρ , where ζ ¯ satisfies div ζ ¯ = 0 , which is called the Hodge vector, and ρ is the Hodge potential of ζ . The main purpose of this article is to initiate a study on the impact of the Hodge vector and its potential on M m . The first result of this article states that a compact Riemannian m -manifold M m is an m -sphere S m ( c ) if and only if (1) for a nonzero constant c , the function − σ / c is a solution of the Poisson equation Δ ρ = m σ , and (2) the Ricci curvature satisfies R i c ζ ¯ , ζ ¯ ≥ φ 2 . The second result states that if M m has constant scalar curvature τ = m ( m − 1 ) c > 0 , then it is an S m ( c ) if and only if the Ricci curvature satisfies R i c ζ ¯ , ζ ¯ ≥ φ 2 and the Hodge potential ρ satisfies a certain static perfect fluid equation. The third result provides another new characterization of S m ( c ) using the affinity tensor of the Hodge vector ζ ¯ of a conformal vector field ζ on a compact Riemannian manifold M m with positive Ricci curvature. The last result states that a complete, connected Riemannian manifold M m , m > 2 , is a Euclidean m -space if and only if it admits a nontrivial conformal vector field ζ whose affinity tensor vanishes identically and ζ annihilates its associated tensor φ .

Keywords: Hodge decomposition; conformal vector field; Hodge vector; affinity tensor; static perfect fluid equation; sphere (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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