General Chen's First Inequalities for Riemannian Submersions and Their Applications

Abstract: In this paper, we introduce and develop the concepts of Chen's first inequalities for Riemannian submersions between Riemannian manifolds. We derive general forms of Chen's first inequalities and analyse their corresponding equality cases. As applications, we apply them to various Riemannian submersions whose total space is real, complex, generalised Sasakian, Sasakian, Kenmotsu, cosymplectic, and $C(α)$ space forms. We construct examples that satisfy the assumptions of the theorems; we observe that equality holds in some examples, while in others it does not.

• The paper establishes Chen’s first inequality for fiber dimensions r>2, relating vertical scalar and sectional curvature to the fibers’ mean curvature and O’Neill tensor.
• The paper derives a second inequality coupling horizontal and vertical curvature through scalar-curvature identities and corrections involving the O’Neill tensors \(\mathcal{T}\) and \(\mathcal{A}\).
• The paper specializes these bounds to real, complex, and generalized Sasakian space forms, with equality attained by totally geodesic fibers such as the Hopf fibration.

Overview and motivation

This paper by Ravindra Singh introduces Chen's first inequality in the setting of Riemannian submersions $F:(M_1^n,g_1)\to (M_2^m,g_2)$, a context in which the inequality had not previously been formulated. Chen's first inequality, originally established for submanifolds of real space forms in 1993 [Chen 1993] and extended to complex space forms shortly thereafter, relates the intrinsic invariant $\delta(2)=\tau-\inf K$ to the squared norm of the mean curvature vector. The author's contribution is twofold: a general inequality along the vertical distribution (the fibers of the submersion), and a second general inequality involving the scalar and sectional curvatures of both horizontal and vertical distributions. Applications are given to submersions whose total space is a real space form, complex space form, or generalized Sasakian space form, with Sasakian, Kenmotsu, cosymplectic, and $C(\alpha)$ space forms obtained as specializations via substitutions into the curvature functions $(c_1,c_2,c_3)$.

The technical machinery follows O'Neill's framework: the fundamental tensors $\mathcal{T}$ and $\mathcal{A}$, the decomposition of $TM_1$ into vertical $\mathcal{V}=\ker F_*$ and horizontal $(\ker F_*)^\perp$ distributions, and O'Neill's curvature equations relating $R^{M_1}$ to the induced curvatures on each distribution.

The vertical inequality

The central result (Theorem 1) states that if the fiber dimension satisfies $\delta(2)=\tau-\inf K$0, then for any 2-plane $\delta(2)=\tau-\inf K$1 spanned by an orthonormal pair $\delta(2)=\tau-\inf K$2,

$\delta(2)=\tau-\inf K$3

where $\delta(2)=\tau-\inf K$4 is the mean curvature vector field of the fibers. A companion result gives the "dual" version involving $\delta(2)=\tau-\inf K$5. The proof combines O'Neill's vertical curvature equation with Chen's algebraic lemma: if real numbers satisfy $\delta(2)=\tau-\inf K$6 with $\delta(2)=\tau-\inf K$7, then $\delta(2)=\tau-\inf K$8, with equality iff $\delta(2)=\tau-\inf K$9.

The equality case is characterized explicitly in terms of components of $C(\alpha)$0: equality holds iff $C(\alpha)$1 for $C(\alpha)$2, off-diagonal terms vanish appropriately, $C(\alpha)$3 for $C(\alpha)$4, and, after a suitable basis choice, $C(\alpha)$5. In particular, totally geodesic fibers ($C(\alpha)$6) always yield equality — consistent with the standard Hopf fibration example below.

Two worked examples illustrate both sides of the coin:

• Strict inequality: a submersion $C(\alpha)$7 between suitably warped metrics where $C(\alpha)$8, violating the equality conditions.
• Equality: a submersion $C(\alpha)$9 with diagonal metric $(c_1,c_2,c_3)$0, for which all $(c_1,c_2,c_3)$1.

Specializations to space forms

Substituting the ambient curvatures into the general theorem yields concrete inequalities. For a real space form $(c_1,c_2,c_3)$2 of constant sectional curvature $(c_1,c_2,c_3)$3:

$(c_1,c_2,c_3)$4

For a complex space form $(c_1,c_2,c_3)$5 of constant holomorphic sectional curvature $(c_1,c_2,c_3)$6, writing $(c_1,c_2,c_3)$7 according to the vertical/horizontal splitting:

$(c_1,c_2,c_3)$8

For a generalized Sasakian space form $(c_1,c_2,c_3)$9, the bound splits into cases depending on whether the structure vector field $\mathcal{T}$0 lies in $\mathcal{T}$1 or $\mathcal{T}$2, with the $\mathcal{T}$3 case carrying an additional correction term $\mathcal{T}$4 where $\mathcal{T}$5. Setting $\mathcal{T}$6 to the standard values recovers Sasakian ($\mathcal{T}$7), Kenmotsu, cosymplectic, and $\mathcal{T}$8 space forms as corollaries.

Notably, the classical Hopf-type fibration $\mathcal{T}$9 with fiber $\mathcal{A}$0 (totally geodesic fibers) attains equality in the real space form case, providing a canonical sharpness witness.

The combined horizontal–vertical inequality

The second main result (Theorem GCFVH) bounds the sum of intrinsic quantities over both distributions from above by their extrinsic counterparts plus O'Neill tensor corrections. With $\mathcal{A}$1 and $\mathcal{A}$2 2-planes:

$\mathcal{A}$3

is bounded above by

$\mathcal{A}$4

The proof derives the scalar curvature identity

$\mathcal{A}$5

and again applies Chen's algebraic lemma; the equality conditions coincide with those of Theorem 1. Corresponding specializations to real, complex, and generalized Sasakian space forms are derived, with explicit curvature data such as $\mathcal{A}$6 in the real case and $\mathcal{A}$7 in the complex case. The same two examples and the Hopf fibration serve to demonstrate both attainment and non-attainment of equality in this setting as well.

Limitations and open questions

Several restrictions are inherent to the results presented. All inequalities require the fiber dimension $\mathcal{A}$8, so the borderline cases $\mathcal{A}$9 remain untreated here. The inequalities are pointwise statements about the vertical (or horizontal-plus-vertical) geometry; no global or topological consequences are drawn. The paper's applications cover only real, complex, and contact-type space forms as total spaces; other ambient geometries (e.g., quaternionic-Kähler, nearly Kähler) treated elsewhere for submanifolds and Riemannian maps are not addressed. The author notes explicitly that extensions of these inequalities to further space forms, and to the settings of submanifolds and Riemannian maps, constitute future work. Additionally, while examples confirm that equality can hold or fail depending on the submersion, no classification of submersions attaining equality is attempted beyond the componentwise characterization of $TM_1$0.

Conclusion

The paper supplies the missing analogue of Chen's first inequality for Riemannian submersions, in two complementary general forms — one confined to the vertical distribution and one coupling horizontal and vertical curvature data through the full scalar curvature decomposition. Both come with complete equality characterizations in terms of O'Neill tensor $TM_1$1, and both specialize cleanly to real, complex, and generalized Sasakian (hence also Sasakian, Kenmotsu, cosymplectic, and $TM_1$2) space forms. Explicit examples, including the Hopf fibration $TM_1$3, verify that the inequalities are sharp in the appropriate sense.