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Pinching Theorems for Statistical Submanifolds in Sasaki-Like Statistical Space Forms
Ali H Alkhaldi1, Mohd Aquib2, Aliya Naaz Siddiqui2
1Department of Mathematics, College of Science, King Khalid University, P.O. Box 9004, Abha 62529, Saudi Arabia.
Entropy (Basel, Switzerland)
|December 3, 2020
Summary
This study establishes upper bounds for specific curvatures on statistical submanifolds within constant curvature Sasaki-like statistical manifolds. It also defines conditions for these manifolds to be η-Einstein and satisfy vacuum Einstein field equations.
Area of Science:
- Differential Geometry
- Mathematical Physics
Background:
- Sasaki-like statistical manifolds are a specialized class of geometric structures.
- Understanding curvature properties is crucial for classifying and analyzing these manifolds.
Purpose of the Study:
- To derive upper bounds for normalized δ-Casorati and generalized normalized δ-Casorati curvatures.
- To investigate the conditions for η-Einstein statistical submanifolds.
- To determine when the metric of these manifolds satisfies vacuum Einstein field equations.
Main Methods:
- Utilizing concepts from differential geometry and curvature theory.
- Analyzing inequalities related to Casorati curvatures.
- Applying conditions for η-Einstein manifolds.
Main Results:
- Established upper bounds for normalized δ-Casorati curvatures.
- Identified the equality cases for these curvature inequalities.
- Provided necessary and sufficient conditions for η-Einstein Sasaki-like statistical manifolds.
- Determined the criteria for the metric to be a solution to vacuum Einstein field equations.
Conclusions:
- The findings contribute to the classification and understanding of Sasaki-like statistical manifolds.
- The results offer insights into the geometric properties and potential physical applications of these manifolds.
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