Generalized Wintgen inequalities for submanifolds of conformally flat manifolds

Abstract: We obtain generalized Wintgen inequalities for submanifolds in conformally flat manifolds. We give some applications for submanifolds in a Riemannian manifold of quasi-constant curvature. Equality cases are also considered.

• The paper establishes a sharp DDVV-type inequality for submanifolds with dimension n ≥ 3 in conformally flat Riemannian manifolds, replacing the constant-curvature term with an explicit ambient Ricci and scalar-curvature correction.
• The paper characterizes equality through canonical shape-operator block forms and shows that minimal submanifolds satisfy a corresponding bound determined solely by ambient Ricci data.
• The paper derives unified corollaries for quasi-constant-curvature manifolds and generalized Robertson–Walker spacetimes, recovering the classical real-space-form and warped-product inequalities as special cases.

From Wintgen's inequality to conformally flat ambients

The generalized Wintgen inequality, originally formulated as the DDVV conjecture of De Smet, Dillen, Verstraelen and Vrancken (2602.08330), asserts that for an isometric immersion $M^n \to \widetilde{M}^m(c)$ into a real space form,

$\rho \leq \|H\|^2 - \rho^\perp + c,$

where $\rho$ and $\rho^\perp$ are the normalized scalar and normal scalar curvatures. The conjecture was resolved in full generality independently by Lu and by Ge–Tang, and has since been adapted to numerous ambient geometries: complex and Sasakian space forms, statistical manifolds, metallic space forms, and warped products. The paper under review extends this program to a substantially broader class of ambient spaces — conformally flat Riemannian manifolds — where no constant-curvature term is available and the ambient curvature enters through its Ricci tensor and scalar curvature.

Preliminaries

For an $n$-dimensional submanifold ($n \geq 3$) of an $m$-dimensional ($m \geq 4$) conformally flat manifold $(\widetilde{M}^m, \widetilde{g})$, the vanishing of the Weyl tensor $\widetilde{C}$ allows the Gauss equation to be rewritten entirely in terms of $\rho \leq \|H\|^2 - \rho^\perp + c,$0, $\rho \leq \|H\|^2 - \rho^\perp + c,$1, and the second fundamental form $\rho \leq \|H\|^2 - \rho^\perp + c,$2. This is the key structural input: in a conformally flat ambient space, the intrinsic curvature of $\rho \leq \|H\|^2 - \rho^\perp + c,$3 is controlled pointwise by the ambient Ricci data plus quadratic terms in $\rho \leq \|H\|^2 - \rho^\perp + c,$4, with explicit coefficients $\rho \leq \|H\|^2 - \rho^\perp + c,$5 and $\rho \leq \|H\|^2 - \rho^\perp + c,$6.

The paper works with two normalizations of normal curvature: the normalized normal scalar curvature $\rho \leq \|H\|^2 - \rho^\perp + c,$7 built from $\rho \leq \|H\|^2 - \rho^\perp + c,$8, and the quantity $\rho \leq \|H\|^2 - \rho^\perp + c,$9, where

$\rho$0

is expressed via commutators of shape operators. These two notions coincide, as shown below, which is what permits passage from the algebraic inequality to the DDVV-type statement.

Main results

Proposition (Wintgen-type inequality with $\rho$1). For any such submanifold,

$\rho$2

The proof combines three ingredients: Mihai's decomposition of $\rho$3 into difference terms and cross terms; Lu's algebraic inequality relating these terms to $\rho$4; and the Gauss equation specialized to conformally flat ambients. Notably, the equality case is fully characterized: equality holds identically if and only if, in suitable orthonormal frames, the shape operators take a canonical block form — $\rho$5 and $\rho$6 differ from scalar multiples of the identity only on a fixed 2-plane spanned by $\rho$7 (with off-diagonal entry $\rho$8), $\rho$9, and $\rho^\perp$0. This mirrors the classical characterization of Wintgen-ideal submanifolds, where the curvature ellipse degenerates appropriately.

Setting $\rho^\perp$1 yields the corresponding bound for minimal submanifolds, in which the right-hand side reduces purely to ambient Ricci data.

Theorem (main result). Using the Ricci equation to identify $\rho^\perp$2 with $\rho^\perp$3, so that $\rho^\perp$4, the proposition upgrades to

$\rho^\perp$5

This genuinely generalizes the sharp DDVV inequality of Ge–Tang and Lu: when the ambient is a real space form of curvature $\rho^\perp$6, the Ricci correction collapses to $\rho^\perp$7 and the classical inequality is recovered. The theorem thus provides a pointwise intrinsic–extrinsic relation valid for all conformally flat ambients, a class far larger than spaces of constant sectional curvature.

Applications to quasi-constant curvature and Robertson–Walker spacetimes

A Riemannian manifold of quasi-constant curvature (Chen–Yano) has curvature tensor of the form $\rho^\perp$8 times the metric combination plus $\rho^\perp$9 times a combination involving a unit vector field $n$0 and its dual 1-form $n$1. Such manifolds are automatically conformally flat, so the main theorem applies directly. Substituting the explicit expressions

$n$2

yields the clean estimate

$n$3

with the analogous version for $n$4. Two special cases follow immediately: if $n$5 is tangent to $n$6, the correction is $n$7; if $n$8 is normal, it is simply $n$9. Taking $n \geq 3$0 recovers the real-space-form case, confirming consistency with the sharp DDVV inequality.

As a further application, a generalized Robertson–Walker spacetime $n \geq 3$1 is itself a manifold of quasi-constant curvature, with associated functions $n \geq 3$2 and $n \geq 3$3. Consequently:

Position of $n \geq 3$4	Inequality
Tangent to $n \geq 3$5	$n \geq 3$6
Normal to $n \geq 3$7	$n \geq 3$8

These reproduce the warped-product DDVV inequality of Roth, now derived as a corollary of the general conformally flat result rather than proved ad hoc. This unification is arguably the main practical value of the paper: inequalities previously established separately for space forms, quasi-constant curvature manifolds, and warped products all become instances of a single theorem.

Limitations and open questions

Several restrictions should be noted. First, the results require $n \geq 3$9 and $m$0; surfaces ($m$1) are excluded, even though the original Wintgen inequality concerns precisely that case, and the paper does not address whether a two-dimensional analogue holds under the same hypotheses. Second, the equality characterization in Proposition 1 is stated for equality holding identically, whereas the corollaries for quasi-constant curvature manifolds assert only pointwise equality at a given $m$2; a global classification of Wintgen-ideal submanifolds in these ambients is not undertaken. Third, the applications rely on the fact that quasi-constant curvature manifolds are conformally flat; the paper leaves open whether comparable inequalities hold for ambient spaces whose Weyl tensor is nonzero but controlled (e.g., nearly quasi-constant curvature, or semisymmetric spaces). Finally, the semi-Riemannian (Lorentzian) analogue — relevant since generalized Robertson–Walker spacetimes are naturally Lorentzian — is not treated here; the framework is strictly Riemannian.

Conclusion

The paper establishes a generalized Wintgen inequality for submanifolds of arbitrary dimension $m$3 and codimension in conformally flat Riemannian manifolds, with the ambient contribution expressed through $m$4 and $m$5, together with a complete shape-operator characterization of the equality case. As corollaries, it recovers the sharp DDVV inequality in real space forms, produces new inequalities for quasi-constant curvature manifolds, and rederives Roth's warped-product inequality for generalized Robertson–Walker spacetimes within a unified framework. The natural open problems are the surface case, the semi-Riemannian extension, and equality classification beyond the pointwise statement.