Quotient Ricci flow as a coupled and weighted super-Ricci flow

Definitions and mathematical statement

The source asks which flows on $M/G$ are induced by Ricci flow on $M$ when $G$ acts freely and isometrically. Existence/uniqueness hypotheses preserving the action and the allowed group are unspecified. The aid treats a smooth invariant unnormalized Ricci flow on a principal torus bundle $T^m\to M^{n+m}\to B^n$, $m\geq1$, in a setting such as closed $M$ where invariance is retained.

Write the invariant metric locally as \[ \overline g=g_{ij}{\,\mathrm{d}} x^i{\,\mathrm{d}} x^j+ G_{ab}({\,\mathrm{d}}\theta^a+A^a_i{\,\mathrm{d}} x^i)({\,\mathrm{d}}\theta^b+A^b_j{\,\mathrm{d}} x^j), \] where $g$ is the quotient metric, $G$ the positive fiber matrix and $A$ the principal connection, with curvature $F^a_{ij}$. The quotient measure obtained from total volume is proportional to $e^{-f}\operatorname{vol}_g$, where $f=-\tfrac12\log\det G$. Put \[ B_i=G^{-1}\partial_iG,\qquad B_i^0=B_i-\frac{\operatorname{tr}B_i}{m}\operatorname{Id}, \] and define the weighted dimension-$(n+m)$ defect tensor by \[ \mathcal D=\tfrac12\partial_tg+\operatorname{Ric}_g+\operatorname{Hess}_g f-\frac1m{\,\mathrm{d}} f\otimes{\,\mathrm{d}} f. \] In these conventions the aid's identity is \[ \mathcal D_{ij}=\frac14\operatorname{tr}(B_i^0B_j^0) +\frac12G_{ab}F^a_{ik}F^b_{j\ell}g^{k\ell}\succeq0. \] Repeated indices are contracted. Each summand is positive semidefinite. Equality as a tensor at a point holds exactly when $F=0$ there and $B_i^0=0$ in every horizontal direction, meaning the horizontal fiber variation is purely conformal. This is the explicitly defined smooth weighted super-Ricci inequality in the aid; convention changes must be tracked.

In the conformally warped flat-fiber sector $G=e^{2u}G_0$ with fixed flat $G_0$ and flat connection, the reported raw quotient/dilaton system is \[ \partial_tg=-2\operatorname{Ric}_g+2m\operatorname{Hess}u+2m{\,\mathrm{d}} u\otimes{\,\mathrm{d}} u, \qquad \partial_tu=\Delta_gu+m|\nabla u|^2. \] Pulling back by the time-dependent diffeomorphisms generated by $-m\nabla u$ removes the Hessian drift and gives List's extended Ricci system \[ \partial_tg=-2\operatorname{Ric}_g+2m{\,\mathrm{d}} u\otimes{\,\mathrm{d}} u, \qquad\partial_tu=\Delta_gu, \] with all fields interpreted after pullback.

For a quotient circle, its length satisfies the exact formula \[ \frac{{\,\mathrm{d}}}{{\,\mathrm{d}} t}\operatorname{Length}(B,g(t)) =m\int_B|\nabla u|^2\,{\,\mathrm{d}} s. \] It is strictly positive whenever the fiber volume, proportional to $e^{mu}$, is nonconstant. One-dimensional ordinary Ricci flow has zero Ricci tensor and constant length, and diffeomorphisms preserve length. Thus even this equality sector need not project to ordinary Ricci flow modulo diffeomorphism. The quotient instead retains essential connection/fiber or dilaton information. The torus identity is a partial answer, not a classification for every free isometric group action.

Short English statement

The supplied torus-bundle calculation yields a coupled quotient metric, connection and fiber flow with an explicit nonnegative weighted Ricci defect. Even its flat conformal-fiber sector gives List's extended flow, and changing circle length obstructs ordinary quotient Ricci flow modulo diffeomorphism.

Sources

• Local UnsolvedMath registry, permanent ID $20003174$, source code catalogue entry; canonical aim-topology-notes.json, notes[261].
• 2007 AIM list Open Problems in Non-negative Sectional Curvature, Section 7, Ricci Flow, Problem 49, workshop Manifolds with nonnegative sectional curvature: https://aimath.org/WWN/nnsectcurvature/nnsectcurvature.pdf. Original remarks and literature field are empty.
• The aid names List's extended Ricci flow but supplies no authoritative full reference or full invariant-bundle evolution theorem.
• Research attempt 1: partial_result, status partially_solved, classification PARTIAL-PROGRESS, rigor proved, novelty confidence low. Its defect identity, equality characterization and length obstruction are machine-generated aid, not peer reviewed, and require independent verification before citation.