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Open AccessArticle
Hodge Decomposition of Conformal Vector Fields on a Riemannian Manifold and Its Applications
by
Hanan Alohali
Hanan Alohali 1
,
Sharief Deshmukh
Sharief Deshmukh 1
,
Bang-Yen Chen
Bang-Yen Chen
Prof. Bang-Yen Chen received his B.S. from Tamkang University in 1965 and his M.Sc. from the Tsing a [...]
Prof. Bang-Yen Chen received his B.S. from Tamkang University in 1965 and his M.Sc. from the National Tsing Hua University in 1967. He obtained his Ph.D. degree from the University of Notre Dame in 1970 under the supervision of Tadashi Nagano. He taught at Tamkang University between 1965 and 1968, and at the National Tsing Hua University in the academic year 1967–1968. After his doctoral years (1968–1970) at the University of Notre Dame, he joined the faculty at Michigan State University as a research associate from 1970 to 1972, where he became an associate professor in 1972, and a full professor in 1976. He was presented with the title of Distinguished Professor in 1990. After 2012, he became a Distinguished Professor Emeritus. He is the author of over 570 works, including 12 books, mainly in differential geometry and related subjects. He was named one of the top 15 famous Taiwanese scientists by SCI Journal.
2,*
and
Hemangi Madhusudan Shah
Hemangi Madhusudan Shah 3
1
Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
2
Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
3
Homi Bhabha National Institute, Harish-Chandra Research Institute, Jhunsi, Allahabad Uttar Pradesh 211019, India
*
Author to whom correspondence should be addressed.
Mathematics 2024, 12(17), 2628; https://doi.org/10.3390/math12172628 (registering DOI)
Submission received: 5 August 2024
/
Revised: 21 August 2024
/
Accepted: 23 August 2024
/
Published: 24 August 2024
Abstract
For a compact Riemannian m-manifold endowed with a nontrivial conformal vector field with a conformal factor , there is an associated skew-symmetric tensor called the associated tensor, and also, admits the Hodge decomposition , where satisfies , which is called the Hodge vector, and is the Hodge potential of . The main purpose of this article is to initiate a study on the impact of the Hodge vector and its potential on . The first result of this article states that a compact Riemannian m-manifold is an m-sphere if and only if (1) for a nonzero constant c, the function is a solution of the Poisson equation , and (2) the Ricci curvature satisfies . The second result states that if has constant scalar curvature , then it is an if and only if the Ricci curvature satisfies and the Hodge potential satisfies a certain static perfect fluid equation. The third result provides another new characterization of using the affinity tensor of the Hodge vector of a conformal vector field on a compact Riemannian manifold with positive Ricci curvature. The last result states that a complete, connected Riemannian manifold , , is a Euclidean m-space if and only if it admits a nontrivial conformal vector field whose affinity tensor vanishes identically and annihilates its associated tensor .
Share and Cite
MDPI and ACS Style
Alohali, H.; Deshmukh, S.; Chen, B.-Y.; Shah, H.M.
Hodge Decomposition of Conformal Vector Fields on a Riemannian Manifold and Its Applications. Mathematics 2024, 12, 2628.
https://doi.org/10.3390/math12172628
AMA Style
Alohali H, Deshmukh S, Chen B-Y, Shah HM.
Hodge Decomposition of Conformal Vector Fields on a Riemannian Manifold and Its Applications. Mathematics. 2024; 12(17):2628.
https://doi.org/10.3390/math12172628
Chicago/Turabian Style
Alohali, Hanan, Sharief Deshmukh, Bang-Yen Chen, and Hemangi Madhusudan Shah.
2024. "Hodge Decomposition of Conformal Vector Fields on a Riemannian Manifold and Its Applications" Mathematics 12, no. 17: 2628.
https://doi.org/10.3390/math12172628
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