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General Casorati inequalities and implications for Riemannian maps and Riemannian submersions

Ravindra Singh, Kiran Meena, Kapish Chand Meena

TL;DR

This work develops a general framework for Casorati inequalities relating normalized horizontal and vertical curvatures to Casorati curvatures for Riemannian maps and Riemannian submersions. It derives two central bounds, $\rho^{\mathcal{H}} \le \delta_C^{\mathcal{H}}(r-1) + \rho^{\mathcal{R}}$ and $\rho^{\mathcal{H}} \le \hat{\delta}_{C}^{\mathcal{H}}(r-1) + \rho^{\mathcal{R}}$, and extends them to specialized ambient spaces including generalized complex and generalized Sasakian space forms. The paper then specializes these inequalities to target/source spaces that are real, complex, real Kähler, Sasakian, Kenmotsu, cosymplectic, and almost $C(\alpha)$ space forms, as well as to invariant/anti-invariant cases, providing explicit formulas involving curvature constants $c_i$ and projections like $\|P^{\mathcal{R}}\|^2$ or $\|P^{\mathcal{H}}\|^2$. Equality cases are characterized by specific structures of the second fundamental form and related tensors, linking the algebraic conditions to geometric characteristics. Overall, the results unify and extend known Casorati inequalities and offer a toolkit for analyzing geometric properties of maps and submersions between diverse space forms.

Abstract

This paper presents general forms of Casorati inequalities for Riemannian maps and Riemannian submersions between Riemannian manifolds. Using these general forms, we obtain Casorati inequalities for Riemannian maps (resp. submersions) whose target (resp. source) spaces are generalized complex and generalized Sasakian space forms. As a consequence, we give Casorati inequalities for Riemannian maps (resp. submersions) when the target (resp. source) spaces are real, complex, real Kähler, Sasakian, Kenmotsu, cosymplectic, and almost $C(α)$ space forms. Toward the affirmation of these general forms, in the particular cases when the target or source spaces are real, complex, Sasakian, and Kenmotsu space forms, we verify known Casorati inequalities for Riemannian maps and Riemannian submersions. Further, we give Casorati inequalities for invariant and anti-invariant Riemannian maps (resp. submersions) whose target (resp. source) spaces are generalized complex and generalized Sasakian space forms. Toward information on geometric characteristics, we discuss the equality cases.

General Casorati inequalities and implications for Riemannian maps and Riemannian submersions

TL;DR

This work develops a general framework for Casorati inequalities relating normalized horizontal and vertical curvatures to Casorati curvatures for Riemannian maps and Riemannian submersions. It derives two central bounds, and , and extends them to specialized ambient spaces including generalized complex and generalized Sasakian space forms. The paper then specializes these inequalities to target/source spaces that are real, complex, real Kähler, Sasakian, Kenmotsu, cosymplectic, and almost space forms, as well as to invariant/anti-invariant cases, providing explicit formulas involving curvature constants and projections like or . Equality cases are characterized by specific structures of the second fundamental form and related tensors, linking the algebraic conditions to geometric characteristics. Overall, the results unify and extend known Casorati inequalities and offer a toolkit for analyzing geometric properties of maps and submersions between diverse space forms.

Abstract

This paper presents general forms of Casorati inequalities for Riemannian maps and Riemannian submersions between Riemannian manifolds. Using these general forms, we obtain Casorati inequalities for Riemannian maps (resp. submersions) whose target (resp. source) spaces are generalized complex and generalized Sasakian space forms. As a consequence, we give Casorati inequalities for Riemannian maps (resp. submersions) when the target (resp. source) spaces are real, complex, real Kähler, Sasakian, Kenmotsu, cosymplectic, and almost space forms. Toward the affirmation of these general forms, in the particular cases when the target or source spaces are real, complex, Sasakian, and Kenmotsu space forms, we verify known Casorati inequalities for Riemannian maps and Riemannian submersions. Further, we give Casorati inequalities for invariant and anti-invariant Riemannian maps (resp. submersions) whose target (resp. source) spaces are generalized complex and generalized Sasakian space forms. Toward information on geometric characteristics, we discuss the equality cases.
Paper Structure (13 sections, 20 theorems, 103 equations)

This paper contains 13 sections, 20 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. General Casorati inequalities and implications for Riemannian maps
  3. General Casorati inequalities for Riemannian maps
  4. Implications for Riemannian maps to generalized complex space forms
  5. Implications for Riemannian maps to generalized Sasakian space forms
  6. General Casorati inequalities and implications for vertical distributions of Riemannian submersions
  7. General Casorati inequalities for vertical distributions of Riemannian submersions
  8. Implications for the vertical distributions of Riemannian submersions from generalized complex space forms
  9. Implications for the vertical distributions of Riemannian submersions from generalized Sasakian space forms
  10. General Casorati inequalities and implications for horizontal distributions of Riemannian submersions
  11. General Casorati inequalities for horizontal distributions of Riemannian submersions
  12. Implications for the horizontal distributions of Riemannian submersions from generalized complex space forms
  13. Implications for the horizontal distributions of Riemannian submersions from generalized Sasakian space forms

Key Result

Lemma 3

Tripathi_2017 Let $\Lambda =\{(z_{1}, \dots, z_{r})\in {\Bbb R}^{r}:z_{1}+\cdots +z_{r}=k\}$ be a hyperplane of ${\Bbb R}^{r}$, and $f:{\Bbb R}^{r}\to {\Bbb R}$ be a quadratic form given by Then the constrained extremum problem $\min \limits_{(z_{1}, \dots, z_{r})\in \Lambda}f$ has the global solution $z_{1} = z_{2} = \cdots = z_{r-1} = \frac{k}{\lambda_1 + 1}, \quad z_{r} = \frac{k}{\lambda_2 +

Theorems & Definitions (37)

  • Definition 1
  • Definition 2
  • Lemma 3
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • Corollary 6
  • Remark 7
  • Theorem 8
  • ...and 27 more