Generalized Chen's inequalities for Riemannian submersions and Riemannian maps with Applications

Abstract: In this paper, we establish generalized B.-Y. Chen inequalities for Riemannian submersions and Riemannian maps between Riemannian manifolds by employing the generalized $δ$-invariants introduced by Chen. We derive optimal inequalities involving the generalized $δ$-invariants associated with the vertical and horizontal distributions together with extrinsic invariants determined by the second fundamental tensors of the submersion and the Riemannian map. Furthermore, we characterize the equality cases through precise algebraic conditions on the corresponding shape operators, providing their geometric interpretation. As applications, we obtain explicit generalized Chen inequalities for Riemannian submersions and Riemannian maps whose total or target manifolds are real space forms and complex space forms. These results extend the classical Chen inequalities as well as several recent Chen-type inequalities for Riemannian submersions and Riemannian maps available in the literature.

• The paper establishes sharp generalized Chen inequalities for the full family of δ- and hat-δ invariants, with explicit coefficients determined by Chen’s algebraic lemma.
• The results provide complete pointwise equality conditions through blockwise constraints on second fundamental tensors, including vanishing off-block terms and prescribed mean-curvature traces.
• The inequalities specialize to real and complex space forms, recover classical results for submersions and maps, and identify open directions involving global rigidity and variable-curvature geometries.

Overview

This paper extends B.-Y. Chen's classical $\delta$-invariant framework to the settings of Riemannian submersions and Riemannian maps between arbitrary Riemannian manifolds. The central contribution is a pair of sharp inequalities relating intrinsic invariants of the vertical (respectively horizontal) distribution to extrinsic invariants determined by the second fundamental tensor of the map, together with complete characterizations of the equality cases. Specializations to real and complex space forms recover and generalize several existing Chen-type inequalities in the literature.

The key technical device is Chen's algebraic lemma: if real numbers $a_1,\ldots,a_k,b$ satisfy $(\sum_i a_i)^2 = (k-1)(\sum_i a_i^2 + b)$ with $k>2$, then $2a_1a_2 \geq b$, with equality iff $a_1+a_2 = a_3 = \cdots = a_k$. The paper applies this lemma after decomposing squared norms of the relevant second fundamental tensors into contributions indexed by mutually orthogonal subspace blocks $D_1,\ldots,D_k$.

Setup for Riemannian submersions

For a Riemannian submersion $\pi:(M_1,g_1)\to(M_2,g_2)$ with $\dim(\ker\pi_\ast)=r$, the tangent bundle splits into vertical ${\cal V}=\ker\pi_\ast$ and horizontal $a_1,\ldots,a_k,b$0 distributions. The geometry is governed by O'Neill's tensors $a_1,\ldots,a_k,b$1 and $a_1,\ldots,a_k,b$2; the mean curvature vector of the fibers is $a_1,\ldots,a_k,b$3 where $a_1,\ldots,a_k,b$4. The paper uses O'Neill's fundamental curvature equations relating $a_1,\ldots,a_k,b$5 restricted to $a_1,\ldots,a_k,b$6 and $a_1,\ldots,a_k,b$7 to the intrinsic curvatures of these distributions plus quadratic terms in $a_1,\ldots,a_k,b$8 and $a_1,\ldots,a_k,b$9.

Scalar curvatures are decomposed as $(\sum_i a_i)^2 = (k-1)(\sum_i a_i^2 + b)$0, $(\sum_i a_i)^2 = (k-1)(\sum_i a_i^2 + b)$1, and mixed terms, with analogous quantities defined on the fiber $(\sum_i a_i)^2 = (k-1)(\sum_i a_i^2 + b)$2 and on $(\sum_i a_i)^2 = (k-1)(\sum_i a_i^2 + b)$3-plane sections $(\sum_i a_i)^2 = (k-1)(\sum_i a_i^2 + b)$4. Curvature tensors of real space forms are written via the Kulkarni–Nomizu product $(\sum_i a_i)^2 = (k-1)(\sum_i a_i^2 + b)$5, and complex space forms via the standard expression involving $(\sum_i a_i)^2 = (k-1)(\sum_i a_i^2 + b)$6.

Main inequality for submersions

Theorem 3.1. For any $(\sum_i a_i)^2 = (k-1)(\sum_i a_i^2 + b)$7 with $(\sum_i a_i)^2 = (k-1)(\sum_i a_i^2 + b)$8, $(\sum_i a_i)^2 = (k-1)(\sum_i a_i^2 + b)$9, $k>2$0,

$k>2$1

where

$k>2$2

Equality holds iff there exist orthonormal bases such that the diagonal components of $k>2$3 along the mean-curvature direction satisfy the block-equality condition $k>2$4, all off-block components vanish, and the sums $k>2$5 vanish for $k>2$6. This is an optimal inequality: the coefficient $k>2$7 is forced by the algebraic lemma, so no larger constant can hold universally.

A companion theorem gives the analogous bound for the "hat" variant $k>2$8, obtained by replacing infima with suprema in the definition of the generalized invariant; the proof reduces to observing that the intrinsic quantities are pointwise constants when optimizing over the choice of orthogonal subspaces.

Two corollaries specialize the result:

• Real space form source: substituting $k>2$9 yields

$2a_1a_2 \geq b$0

• Complex space form source: the bound acquires Kähler correction terms,

$2a_1a_2 \geq b$1

where $2a_1a_2 \geq b$2 is the vertical component of $2a_1a_2 \geq b$3 and $2a_1a_2 \geq b$4. These reproduce, at $2a_1a_2 \geq b$5, $2a_1a_2 \geq b$6, the classical Chen inequalities previously established for submersions.

Extension to Riemannian maps

The second half develops the parallel theory for Riemannian maps $2a_1a_2 \geq b$7 with $2a_1a_2 \geq b$8, which interpolate between immersions and submersions. Here the splitting is $2a_1a_2 \geq b$9 and $a_1+a_2 = a_3 = \cdots = a_k$0, with $a_1+a_2 = a_3 = \cdots = a_k$1. The extrinsic invariant is now the norm of the trace of the second fundamental form $a_1+a_2 = a_3 = \cdots = a_k$2, whose Gauss equation reads

$a_1+a_2 = a_3 = \cdots = a_k$3

Main theorem. For any admissible tuple,

$a_1+a_2 = a_3 = \cdots = a_k$4

Note the direction of the inequality reverses relative to the submersion case, reflecting the sign structure of the Gauss equation for maps versus that of O'Neill's equations. The equality characterization is structurally identical: block constancy of the diagonal coefficients $a_1+a_2 = a_3 = \cdots = a_k$5 along the direction of $a_1+a_2 = a_3 = \cdots = a_k$6, vanishing of off-block components, and vanishing of block traces in the remaining normal directions.

Specializing the target to a real space form gives

$a_1+a_2 = a_3 = \cdots = a_k$7

and to a complex space form gives the corresponding bound with the $a_1+a_2 = a_3 = \cdots = a_k$8 and $a_1+a_2 = a_3 = \cdots = a_k$9 correction terms involving the horizontal component of $D_1,\ldots,D_k$0 composed with $D_1,\ldots,D_k$1. At $D_1,\ldots,D_k$2, $D_1,\ldots,D_k$3, these reduce to Şahin's Chen first inequality for Riemannian maps and its complex-space-form analogue due to Meena, Şahin, and Shah.

Limitations and open questions

Several points qualify the results. First, the inequalities are pointwise statements; the paper does not address global consequences such as rigidity or classification of maps attaining equality on all of $D_1,\ldots,D_k$4, which in the classical submanifold setting constitute a substantial literature. Second, the applications cover only constant-curvature sources/targets (real and complex space forms); extensions to generalized Sasakian or generalized complex space forms—treated elsewhere by the author for the non-generalized inequalities—are not carried out here. Third, the equality analysis presumes the mean curvature vector (or $D_1,\ldots,D_k$5) lies along a single chosen normal direction, which is always achievable by basis rotation but means the stated conditions are basis-dependent in form. Finally, the paper does not exhibit concrete examples of maps attaining equality, so the sharpness of the bounds, while guaranteed by the algebraic lemma, is not illustrated geometrically.

Conclusion

The paper unifies and generalizes the known Chen-type inequalities for Riemannian submersions and Riemannian maps by working with the full family of generalized $D_1,\ldots,D_k$6-invariants $D_1,\ldots,D_k$7 and $D_1,\ldots,D_k$8 rather than only the original $D_1,\ldots,D_k$9. The resulting inequalities are sharp, come with explicit algebraic equality conditions on the shape operators, and specialize cleanly to real and complex space forms, recovering prior results as special cases. The natural open problems are the global rigidity theory of equality cases and extensions to ambient spaces with non-constant curvature.