Disaffinity Vectors on a Riemannian Manifold and Their Applications

Disaffinity Vectors on a Riemannian Manifold and Their Applications

A disaffinity vector on a Riemannian manifold <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>M</mi><mo>,</mo><mi>g</mi><mo>)</mo>&...

Full description

Saved in:
Bibliographic Details
Main Authors: Sharief Deshmukh, Amira Ishan, Bang-Yen Chen
Format: Article
Language:English
Published: MDPI AG 2024-11-01
Series:Mathematics
Subjects:
affinity tensor
disaffinity function
disaffinity vector
Ricci soliton
Yamabe soliton
isometric to Euclidean space
Online Access:https://www.mdpi.com/2227-7390/12/23/3659
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:A disaffinity vector on a Riemannian manifold <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>M</mi><mo>,</mo><mi>g</mi><mo>)</mo></mrow></semantics></math></inline-formula> is a vector field whose affinity tensor vanishes. In this paper, we observe that nontrivial disaffinity functions offer obstruction to the topology of <i>M</i> and show that the existence of a nontrivial disaffinity function on <i>M</i> does not allow <i>M</i> to be compact. A characterization of the Euclidean space is also obtained by using nontrivial disaffinity functions. Further, we study properties of disaffinity vectors on <i>M</i> and prove that every Killing vector field is a disaffinity vector. Then, we prove that if the potential field <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ζ</mi></semantics></math></inline-formula> of a Ricci soliton <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="(" close=")"><mi>M</mi><mo>,</mo><mi>g</mi><mo>,</mo><mi>ζ</mi><mo>,</mo><mi>λ</mi></mfenced></semantics></math></inline-formula> is a disaffinity vector, then the scalar curvature is constant. As an application, we obtain conditions under which a Ricci soliton <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="(" close=")"><mi>M</mi><mo>,</mo><mi>g</mi><mo>,</mo><mi>ζ</mi><mo>,</mo><mi>λ</mi></mfenced></semantics></math></inline-formula> is trivial. Finally, we prove that a Yamabe soliton <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="(" close=")"><mi>M</mi><mo>,</mo><mi>g</mi><mo>,</mo><mi>ξ</mi><mo>,</mo><mi>λ</mi></mfenced></semantics></math></inline-formula> with a disaffinity potential field <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ξ</mi></semantics></math></inline-formula> is trivial.
ISSN:2227-7390

Similar Items