Ricci Curvature Inequalities for Skew CR-Warped Product Submanifolds in Complex Space Forms
Abstract
:1. Introduction
- 1.
- For each unit tangent vector we have
- 2.
- If , then the unit tangent vector χ at p satisfies the equality case of (1) if and only if χ lies in the relative null space at
- 3.
- The equality case holds identically for all unit tangent vectors at x if and only if either p is a totally geodesic point or and p is a totally umbilical point.
2. Preliminaries
- (i)
- Each is a distinct eigenvalue of with
- (ii)
- The distributions of and are independent of
3. Skew CR-Warped Product Submanifolds
4. Ricci Curvature for Skew CR-Warped Product Submanifold
- (1)
- The Ricci curvature satisfies the following expressions:
- (i)
- If , then
- (ii)
- , then
- (iii)
- If , then
- (2)
- If for each point , then there is a unit vector field χ which satisfies the equality of (1) iff is mixed totally geodesic and at x.
- (3)
- For the equality case we have
- (a)
- The equality of (23) holds identically for all unit vector fields tangential to at each iff is mixed TG and totally geodesic SCR W-P submanifold in .
- (b)
- The equality of (24) holds identically for all unit vector fields tangential to at each iff S is mixed totally geodesic and either is - totally geodesic SCR W-P submanifold or is a totally umbilical in with dim .
- (c)
- The equality of (25) holds identically for all unit vector fields tangential to at each iff S is mixed totally geodesic and either is - totally geodesic SCR W-P or is a totally umbilical in with dim .
- (d)
- The equality case of (1) holds identically for all unit tangent vectors to at each iff either is totally geodesic submanifold or is a mixed totally geodesic totally umbilical and L totally geodesic submanifold with dim and dim
- (1)
- The Ricci curvature satisfy the following inequalities
- (i)
- If , then
- (ii)
- If , then
- (2)
- If , then each point there is a unit vector field χ which satisfies the equality case of (1) if and only if is mixed totally geodesic and χ lies in the relative null space at x.
- (3)
- For the equality case we have
- (a)
- The equality case of (56) holds identically for all unit vector fields tangent to at each iff is mixed totally geodesic and totally geodesic CR W-P submanifold in .
- (b)
- The equality case of (57) holds identically for all unit vector fields tangent to at each iff is mixed totally geodesic and either is - totally geodesic CR-warped product or is a totally umbilical in with dim .
- (c)
- The equality case of (1) holds identically for all unit tangent vectors to at each if and only if either is totally geodesic submanifold or is a mixed totally geodesic totally umbilical and totally geodesic submanifold with dim
- (1)
- The Ricci curvature satisfy the following inequalities
- (i)
- If , then
- (ii)
- , then
- (iii)
- If , then
- (2)
- If for each point , then there is a unit vector field χ which satisfies the equality of (1) iff is mixed totally geodesic and at x.
- (3)
- For the equality case we have
- (a)
- The equality of (58) holds identically for all unit vector fields tangent to at each iff is mixed TG and totally geodesic SCR W-P submanifold in .
- (b)
- The equality of (59) holds identically for all unit vector fields tangent to at each iff S is mixed totally geodesic and either is - totally geodesic SCR W-P submanifold or is a totally umbilical in with dim .
- (c)
- The equality of (60) holds identically for all unit vector fields tangent to at each iff S is mixed totally geodesic and either is - totally geodesic SCR W-P or is a totally umbilical in with dim .
- (d)
- The equality case of (1) holds identically for all unit tangent vectors to at each iff either is totally geodesic submanifold or is a mixed totally geodesic totally umbilical and L totally geodesic submanifold with dim and dim
5. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Abbreviations
W-P | Warped product |
W-F | Warping function |
CSF | Complex Space form |
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Ali Khan, M.; Aldayel, I. Ricci Curvature Inequalities for Skew CR-Warped Product Submanifolds in Complex Space Forms. Mathematics 2020, 8, 1317. https://doi.org/10.3390/math8081317
Ali Khan M, Aldayel I. Ricci Curvature Inequalities for Skew CR-Warped Product Submanifolds in Complex Space Forms. Mathematics. 2020; 8(8):1317. https://doi.org/10.3390/math8081317
Chicago/Turabian StyleAli Khan, Meraj, and Ibrahim Aldayel. 2020. "Ricci Curvature Inequalities for Skew CR-Warped Product Submanifolds in Complex Space Forms" Mathematics 8, no. 8: 1317. https://doi.org/10.3390/math8081317