Normal Mean Ricci Curvature

• Normal mean Ricci curvature is the average of mixed sectional curvatures between a chosen d-plane and its orthogonal complement in a Riemannian manifold.
• It governs small normal-sphere volume expansions and appears as the r²/6 coefficient in geodesic sphere asymptotics, linking local geometry to curvature averages.
• The Bochner–Weitzenböck identity for simple d-vectors, involving normal mean Ricci, yields eigenvalue estimates and Lichnerowicz-type lower bounds on closed manifolds.

Normal mean Ricci curvature is a pointwise subspace average of sectional curvature attached to a $d$-dimensional plane $\Pi\subset T_pM$ in a Riemannian $n$-manifold. It is defined by averaging the sectional curvatures of all $2$-planes spanned by one direction in $\Pi$ and one direction in $\Pi^\perp$. In "Intrinsic and Normal Mean Ricci Curvatures: A Bochner--Weitzenboeck Identity for Simple d-Vectors" the notion is introduced together with its companion, the intrinsic mean Ricci, and is shown to govern both small normal-sphere volume expansions and a Bochner--Weitzenböck identity for simple $d$-vectors (Gajer et al., 14 Aug 2025).

1. Definition and geometric content

Let $(M^n,g)$ be a Riemannian $n$-manifold, let $p\in M$, and let

$\Pi\subset T_pM$0

Choose orthonormal bases

$\Pi\subset T_pM$1

For each mixed $\Pi\subset T_pM$2-plane $\Pi\subset T_pM$3, its sectional curvature is

$\Pi\subset T_pM$4

The normal mean Ricci curvature of $\Pi\subset T_pM$5 is then

$\Pi\subset T_pM$6

This quantity is independent of the choice of orthonormal frames and depends only on the splitting $\Pi\subset T_pM$7 (Gajer et al., 14 Aug 2025).

The defining average is neither the ordinary Ricci curvature of a vector nor the scalar curvature of a subspace. It is a subspace invariant built from all mixed sectional curvatures between $\Pi\subset T_pM$8 and its orthogonal complement. In the terminology of the same work, it is the "normal (mixed) mean Ricci" of $\Pi\subset T_pM$9, whereas the intrinsic mean Ricci is the average of sectional curvatures of $n$0-planes contained in $n$1 (Gajer et al., 14 Aug 2025).

2. Jacobi-field interpretation and local volume asymptotics

A central feature of normal mean Ricci is its appearance in small-radius geometry transverse to $n$2. Fix $n$3 and consider the normal sphere of radius $n$4,

$n$5

A Jacobi-field calculation yields a small-$n$6 expansion for the $n$7-dimensional volume element: $n$8 Accordingly, the normal mean Ricci occurs as the $n$9 coefficient in the transverse sphere-volume expansion (Gajer et al., 14 Aug 2025).

The same paper places this result alongside the corresponding intrinsic construction: intrinsic mean Ricci appears as the $2$0 coefficient in the intrinsic $2$1-sphere volume element, while normal mean Ricci appears in the normal $2$2-sphere volume element. This gives the invariant a direct metric interpretation. Rather than being introduced purely algebraically, it is extracted from the second-order behavior of geodesic spheres in directions orthogonal to $2$3 (Gajer et al., 14 Aug 2025).

A plausible implication is that normal mean Ricci plays, for transverse volume comparison relative to a chosen $2$4-plane, a role analogous to that played by ordinary Ricci curvature in classical geodesic-ball asymptotics. The paper states this in the more specific form that the invariant "appears in small-sphere/tube expansions transverse to $2$5" (Gajer et al., 14 Aug 2025).

3. Bochner--Weitzenböck identity for simple $2$6-vectors

The principal analytic result is a Bochner--Weitzenböck identity specialized to simple $2$7-vectors. On the bundle $2$8, the Hodge (Lichnerowicz) Laplacian is

$2$9

where $\Pi$0 is the curvature endomorphism on $\Pi$1. If

$\Pi$2

is a pointwise orthonormal simple $\Pi$3-vector spanning $\Pi$4, then

$\Pi$5

Thus the curvature term in the Bochner formula is exactly $\Pi$6 times the normal mean Ricci of the underlying $\Pi$7-plane (Gajer et al., 14 Aug 2025).

For a unit simple field $\Pi$8, the usual Bochner identity becomes

$\Pi$9

This formula is structurally notable because the curvature term collapses from the full endomorphism $\Pi^\perp$0 to a scalar quantity determined by the mixed sectional curvatures between $\Pi^\perp$1 and $\Pi^\perp$2 (Gajer et al., 14 Aug 2025).

The identity applies specifically to simple $\Pi^\perp$3-vectors, not arbitrary $\Pi^\perp$4-forms or decomposable tensors in a broader sense. That restriction is essential: the simplification of the curvature term depends on the presence of an orthonormal frame $\Pi^\perp$5 spanning a genuine $\Pi^\perp$6-plane $\Pi^\perp$7.

4. Vanishing and eigenvalue estimates on closed manifolds

The paper defines the global lower bound

$\Pi^\perp$8

This quantity packages the infimum of the pointwise normal mean Ricci over the Grassmann bundle of $\Pi^\perp$9-planes (Gajer et al., 14 Aug 2025).

Two analytic consequences are immediate on a closed manifold.

First, if $d$0, then any smooth simple $d$1-vector field $d$2 satisfying $d$3 must vanish identically. The argument integrates the Bochner identity and uses positivity of the curvature term to force $d$4 (Gajer et al., 14 Aug 2025).

Second, if $d$5 is a nonzero smooth simple $d$6-vector field with

$d$7

then

$d$8

Equivalently,

$d$9

This is presented as a Lichnerowicz-type lower bound for the first eigenvalue of the Hodge Laplacian on simple $(M^n,g)$0-eigenfields (Gajer et al., 14 Aug 2025).

These consequences place normal mean Ricci within the Bochner tradition, but with a domain of application adapted to decomposable multivector fields. The positivity hypothesis is not a lower bound on ordinary Ricci curvature, nor on the full curvature operator on $(M^n,g)$1; it is a lower bound on the mixed average associated with each $(M^n,g)$2-plane.

5. Explicit formulas and model spaces

The paper gives explicit computations of $(M^n,g)$3 in several standard geometries (Gajer et al., 14 Aug 2025).

Geometry	Condition on $(M^n,g)$4	$(M^n,g)$5
Space form $(M^n,g)$6 of constant sectional curvature $(M^n,g)$7	Any $(M^n,g)$8-plane	$(M^n,g)$9
$n$0 with Fubini–Study normalization (holomorphic sectional curvature $n$1)	$n$2 a $n$3-invariant real $n$4-plane	$n$5
$n$6 with Fubini–Study normalization	$n$7 totally real of dimension $n$8	$n$9
$p\in M$0 with $p\in M$1, $p\in M$2	$p\in M$3, $p\in M$4, $p\in M$5	$p\in M$6

In space forms, the invariant is constant and equals the ambient sectional curvature. Consequently,

$p\in M$7

for every simple $p\in M$8-vector $p\in M$9 (Gajer et al., 14 Aug 2025).

For complex projective space, the formulas distinguish $\Pi\subset T_pM$00-invariant and totally real planes. This shows that normal mean Ricci is sensitive not only to the ambient metric but also to how the chosen $\Pi\subset T_pM$01-plane sits inside the tangent space relative to the complex structure (Gajer et al., 14 Aug 2025).

For Riemannian products, the explicit formula records how the average splits according to the decomposition $\Pi\subset T_pM$02. The same work states that warped products, the Heisenberg group, and surfaces of revolution admit similar explicit formulae for $\Pi\subset T_pM$03 in terms of base/fiber curvature and warping data; in all cases, the invariant measures the average "bending" of all $\Pi\subset T_pM$04-planes that meet $\Pi\subset T_pM$05 orthogonally (Gajer et al., 14 Aug 2025).

6. Relation to Ricci pinching, normal curvature, and adjacent frameworks

The terminology surrounding "Ricci," "mean curvature," and "normal curvature" is not uniform across the literature, and normal mean Ricci occupies a distinct place within it.

In the framework of compact submanifolds in space forms, one encounters lower bounds of the form

$\Pi\subset T_pM$06

used to derive sphere theorems and rigidity without assuming $\Pi\subset T_pM$07 (Dajczer et al., 2023). In the hypersurface case in $\Pi\subset T_pM$08, a different pinching condition,

$\Pi\subset T_pM$09

is combined with a Bochner--Weitzenböck formula on $\Pi\subset T_pM$10-forms to obtain topological consequences and Clifford-torus rigidity (Dajczer et al., 2023). These are intrinsic Ricci lower bounds expressed in terms of mean curvature.

By contrast, in the theory of real hypersurfaces in non-flat complex space forms, "normal curvature in direction $\Pi\subset T_pM$11" denotes

$\Pi\subset T_pM$12

and appears in the inequality

$\Pi\subset T_pM$13

(Sasahara, 2018). In submanifolds of Bochner–Kähler manifolds, Chen-type inequalities again relate $\Pi\subset T_pM$14 to $\Pi\subset T_pM$15 and ambient curvature terms (Lone et al., 2016).

Normal mean Ricci is different from all of these quantities. It is not the Ricci tensor of the submanifold, not the scalar mean curvature of an immersion, and not the hypersurface normal curvature $\Pi\subset T_pM$16. It is instead the average of mixed sectional curvatures determined by a splitting $\Pi\subset T_pM$17 (Gajer et al., 14 Aug 2025). This distinction suggests that the invariant belongs to the intrinsic Riemannian geometry of tangent-space decompositions, while still interfacing naturally with Bochner theory and with volume-comparison phenomena in directions normal to $\Pi\subset T_pM$18.

Within that broader landscape, the main significance of normal mean Ricci is that it isolates a curvature average precisely adapted to simple $\Pi\subset T_pM$19-vectors. The resulting Bochner identity, vanishing criterion, and eigenvalue lower bound indicate that mixed sectional curvature averages can play the same structural role for decomposable multivector fields that ordinary Ricci curvature plays for vector fields and classical differential forms (Gajer et al., 14 Aug 2025).