Positive TriRic Curvature in Geometry

• Positive triRic curvature is a geometric property of Riemannian manifolds defined by summing sectional curvatures over selected pairs in an orthonormal basis, serving as an intermediate condition between Ricci and scalar curvatures.
• On homogeneous spaces like the Wallach flag manifold, explicit formulas and parameterizations determine regions (e.g., R3) where metrics exhibit positive triRic curvature, facilitating concrete computational analysis.
• This curvature condition is pivotal in understanding the behavior under Ricci flow, enabling controlled gluing constructions and establishing sharp homological systolic inequalities that reveal rigidity and topological constraints.

Positive triRic curvature is a geometric property of Riemannian manifolds that lies between positivity of the Ricci and scalar curvatures, representing a form of “intermediate” curvature condition. The associated invariants, existence criteria, behavior under geometric flows, and topological consequences have been subject to substantial recent investigation, particularly in connection with rigidity, gluing constructions, and homological systolic inequalities.

1. Definition and Characterizations

Positive triRic curvature is a specific instance of the general concept of $m$-intermediate curvature on an $n$-dimensional Riemannian manifold $(M^n, g)$. Given an orthonormal basis $\{e_i\}_{i=1}^n$ of $T_pM$ at a point $p \in M$, the $m$-intermediate curvature is defined as the sum of the sectional curvatures over all $m$-element subsets: $\sum_{p=1}^m \sum_{q=p+1}^n R(e_p, e_q, e_p, e_q)$ where $R$ is the Riemannian curvature tensor. For $m = 2$, this is termed triRic curvature, and a Riemannian metric $g$ is said to have positive triRic curvature if

$$\sum_{p=1}^2 \sum_{q=p+1}^n R(e_p, e_q, e_p, e_q) > 0$$

for every orthonormal basis at every point (Chow et al., 2023). This notion interpolates between positive Ricci curvature ($m=1$) and positive scalar curvature ($m=n-1$), connecting several classical curvature conditions.

An alternative, tensorial definition for positive triRic curvature, especially in higher dimensions, employs the “$\gamma$-triRic” operator: $$\gamma\operatorname{-triRic}(e_i,e_j,e_l) = \gamma\,\operatorname{Ric}(e_i,e_i) + \gamma\text{-}\operatorname{biRic}_{e_i^\perp}(e_j,e_l)$$ where the partial biRicci curvature is

$$\gamma\text{-}\operatorname{biRic}_{e_i^\perp}(e_j,e_l) = \gamma \sum_{m \neq i} R_{m j m j} + \sum_{m \neq i} R_{m l m l} - \gamma R_{j l j l}$$

and a manifold has positive triRic curvature if the infimum of this expression over all orthonormal triples $(e_i, e_j, e_l)$ is strictly positive (Chen et al., 18 Jan 2026).

2. Explicit Formulas and Computations on Homogeneous Spaces

On the homogeneous Wallach flag manifold $M = \mathrm{SU}(3)/\mathrm{T}^2$, every invariant Riemannian metric of fixed volume can be parametrized as $g = (x, y, z)$ with $x, y, z > 0$ and $x + y + z = 1$. Sectional curvatures in the canonical basis have explicit forms:

• $K(X_1,X_2)=1/x$, $K(X_3,X_4)=1/y$, $K(X_5,X_6)=1/z$.
• Mixed curvatures involve rational expressions in $x$, $y$, $z$.

The triRic curvature $\rho_3(x,y,z)$ is computed by taking minima over all possible 3-term sums of the relevant sectional curvatures. This is encoded in a system of inequalities $R_{ij}^{a,b,c}(x,y) > 0$ (for various multi-indices $(a, b, c)$) over the simplex $x, y > 0$, $x + y < 1$, leading to a well-defined region $R_3$ of metrics with positive triRic curvature (Cavenaghi et al., 2023).

Parameterization	Metric Region	TriRic Formula
$(x, y, z)$	$x, y, z > 0$, $x + y + z = 1$	See explicit $K(X_i,X_j)$ sums
$R_3 \subset T$	$R_{ij}^{a,b,c}(x,y) > 0$, five per block	Curvilinear hexagon in Wallach triangle

3. Evolution Under Ricci Flow

The preservation of positive triRic curvature under the Ricci flow has been explicitly analyzed in the setting of homogeneous manifolds. Restricting to $\mathrm{SU}(3)/\mathrm{T}^2$, the Ricci flow induces a two-dimensional dynamical system for $(x, y)$, preserving the volume: $x' = u(x, y),\quad y' = v(x, y)$ with concrete polynomial formulae for $u$ and $v$. Most orbits beginning inside $R_3$ (the region of positive triRic curvature) exit in finite forward time. Only three heteroclinic trajectories, corresponding to one-parameter families of invariant metrics, remain entirely within $R_3$ for all $t$; these represent the only eternal positive triRic metrics under flow. Thus, positive triRic curvature is not generally a Ricci-flow-invariant condition (Cavenaghi et al., 2023).

4. Gluing, Obstructions, and Constructions

Positive triRic curvature satisfies a local gluing property under controlled variations of the metric near a boundary. If two Riemannian metrics $g$, $\bar{g}$ with positive $m$-intermediate curvature coincide on the boundary and their respective second fundamental forms satisfy a strict $m$-positivity condition, then there exists a smooth interpolating family of metrics with positive $m$-intermediate curvature between them.

However, positive intermediate curvature (including the triRic case, $m=2$) is obstructed in the presence of certain topological structures. On a "partial torical band" $N = M^{n-2} \times S^1 \times [-1,1]$, no metric can admit both positive triRic curvature in the interior and strictly 2-convex boundary. This result extends the classical Gromov–Lawson obstructions for positive scalar curvature to the intermediate/triRic regime (Chow et al., 2023).

5. Systolic Geometry and Rigidity

A central application of positive triRic curvature concerns homological systolic inequalities in dimensions $n \geq 4$. For manifolds $(M^n, g)$ with $H_{n-2}(M) \ne 0$ and $$\inf_{p \in M} \operatorname{triRic}_g(p) \geq n-3>0$$, the $(n-2)$-systole (smallest area of a nontrivial $(n-2)$-cycle),

$\operatorname{sys}_{n-2}(M, g)$

satisfies the sharp inequality: $$\operatorname{sys}_{n-2}(M, g) \cdot \bigl( \inf \operatorname{triRic}_g \bigr)^{\frac{n-2}{2}} \le (n-3)^{\frac{n-2}{2}} |S^{n-2}|$$ with equality only if $M$ is covered by the Riemannian product $S^{n-2} \times \mathbb{R}^2$ with the standard metric (Chen et al., 18 Jan 2026).

The proof utilizes stable weighted $k$-slicings (iterated area-minimizing subsurfaces with weights from nontrivial cup product cohomology classes) and a refined Bishop–Gromov type weighted volume comparison theorem. Rigidity is established by metric-deformation arguments forcing a global geometric splitting.

6. Connections to Classical Curvatures and Models

The triRic condition bridges the classical scalar and Ricci curvature conditions. For constant curvature manifolds $(M^n, g)$ with sectional curvature $K > 0$, one finds $\operatorname{triRic}(p) = (n-2)K$. For products $S^{n-2}(r) \times \mathbb{R}^2$, $\operatorname{triRic} \equiv n-3$, precisely characterizing the equality cases in systolic inequalities. When $n=3$, $\operatorname{triRic}$ specializes to scalar curvature, recovering the well-known scalar curvature systolic rigidity for $S^2 \times \mathbb{R}$ (Chen et al., 18 Jan 2026).

7. Topological Constraints

Positive triRic curvature imposes strong topological restrictions analogous to those known for positive scalar and Ricci curvatures but is sensitive to finer aspects of cohomology and higher codimension cycles. For instance, any metric with positive TriRic curvature on $M^{n-2} \times S^1 \times [-1,1]$ cannot have strictly 2-convex boundaries, reflecting a generalized rigidity and non-existence phenomenon for partial torical bands. This positions triRic curvature as a fundamental intermediary in the classification and analysis of Riemannian manifolds with positive curvature constraints (Chow et al., 2023).