Ricci Solitons in Finsler Geometry

• Ricci solitons are self-similar solutions to the Ricci flow, defined in both Riemannian and Finsler settings, that generalize Einstein metrics and serve as singularity models.
• They involve the evolution of metrics via Finsler–Ricci flow using curvature tensors and Lie derivatives, linking geometric analysis with global topological properties.
• The theory establishes finiteness theorems, analogous to Myers’ theorem, which imply finite fundamental groups and drive further research into canonical metric deformations.

A Ricci soliton is a self-similar solution to the Ricci flow, generalizing Einstein metrics and serving as singularity models for the flow equation. The theory extends from Riemannian and pseudo-Riemannian geometry to Finsler spaces, linking canonical metric deformations with global topological and analytic properties. In Finsler geometry—a generalization of Riemannian geometry where the quadratic restriction on the norm is relaxed—Ricci solitons are defined via the Finsler–Ricci flow and associated curvature tensors, providing fundamental links between geometric analysis, topology, and metric geometry (Bidabad et al., 2018).

1. Finsler Metrics and the Ricci Tensor

Given a smooth $n$-manifold $M$, a Finsler structure is a function $F : TM \to [0, \infty)$ such that:

• $F$ is $C^\infty$ on $TM \setminus \{0\}$,
• $F(x,\lambda y) = \lambda F(x,y)$ for all $\lambda > 0$,
• The fundamental tensor $$g_{ij}(x,y) = \frac{1}{2} \frac{\partial^2[F^2(x,y)]}{\partial y^i \partial y^j}$$ is positive definite for $(x,y) \in TM \setminus \{0\}$.

The canonical Cartan (or Chern) connection enables the definition of the hh-curvature tensor $R^i{}_k$ in terms of the spray coefficients $G^i$. The Ricci scalar is defined as $\mathrm{Ric}(x,y) = R^i{}_i(x,y)$, and one can further define a Ricci-type tensor: $$\mathrm{Ric}_{ij}(x,y) := \frac{1}{2} \frac{\partial^2}{\partial y^i \partial y^j}[F^2(x,y)\,\mathrm{Ric}(x,y)].$$ This structure extends the algebraic and analytic features of Riemannian curvature to the much larger Finsler category (Bidabad et al., 2018).

2. Finsler–Ricci Flow and Ricci Soliton Equation

The Finsler–Ricci flow evolves the metric as: $$\frac{\partial}{\partial t} g_{ij} = -2\,\mathrm{Ric}_{ij}.$$ A Ricci soliton is a self-similar solution, i.e., a metric $g_{ij}(t)$ evolving by diffeomorphisms and scaling: $$\frac{\partial}{\partial t} g_{ij} = -2\,\mathrm{Ric}_{ij} + \mathcal{L}_V g_{ij},$$ for some vector field $V$ on $M$. Contracting with $y^i y^j$ yields the pointwise equation

$$2F^2\,\mathrm{Ric} + \mathcal{L}_V(F^2) = 2\lambda F^2,$$

where $\lambda \in \mathbb{R}$ encodes the nature of the soliton: shrinking $(\lambda>0)$, steady $(\lambda=0)$, or expanding $(\lambda<0)$. This framework generalizes the classic soliton equation in the Riemannian case, adjusting both the Ricci and the Lie derivative terms according to the Finsler geometric structure (Bidabad et al., 2018).

3. Topological and Classification Results: The Finiteness Theorem

Shrinking Ricci solitons in Finsler geometry mirror Myers-type theorems from Riemannian geometry. The principal result states:

Theorem (Bidabad et al., 2018).

Let $(M, F, V)$ be a forward (or backward) geodesically complete shrinking Finsler–Ricci soliton, i.e.,

$$2F^2\,\mathrm{Ric} + \mathcal{L}_V(F^2) = 2\lambda F^2,\quad \lambda>0.$$

Then the fundamental group $\pi_1(M)$ is finite, and hence $H^1_{\mathrm{dR}}(M) = 0$.

The proof combines:

• A sharp distance estimate based on second variation of Finsler arc-length, bounding the integral of the Ricci scalar along minimal geodesics in terms of local maxima of $|\mathrm{Ric}(x,y)|$ and the dimension,
• A comparison between the soliton inequality and Cauchy–Schwarz bounds on boundary terms in the Lie derivative, producing a uniform upper bound on distances between points and their deck transforms,
• The conclusion that the universal cover has only finitely many deck transformations (since they all lie within a bounded discrete set) (Bidabad et al., 2018).

4. Explicit Constructions and Known Examples

Explicit non-Riemannian Finsler–Ricci solitons are rare. Known examples include:

• Randers-type metrics (solutions to $$\mathrm{Ric}_{ij} + \frac{1}{2}(\nabla_i W_j + \nabla_j W_i) = \lambda g_{ij}$$), where $W$ is a conformal vector field,
• Metrics arising from Zermelo navigation on Einstein Riemannian spaces.

These examples yield genuine shrinking and expanding Finslerian solitons with no Riemannian analog, thus expanding the landscape of canonical metric geometries on non-Riemannian spaces (Bidabad et al., 2018).

5. Ricci Solitons vs. Einstein Metrics in Finsler Geometry

Ricci solitons generalize Einstein metrics in the Finsler category, just as in Riemannian geometry:

• For $\lambda = 0$ and $V$ Killing, one recovers Finsler–Einstein metrics,
• Shrinking and expanding solitons correspond to canonical self-similar solutions to Finsler–Ricci flow.

The finiteness theorem above is a direct analog of Myers' theorem and Wylie's result for Riemannian shrinkers, affirming that Finsler–Ricci flow imposes comparable topological constraints and opens avenues for the study of singularity formation, long-time behavior, and the structure of canonical metrics in Finsler geometry (Bidabad et al., 2018).

6. Research Directions and Significance

Finslerian Ricci solitons unify geometric analysis and the topology of non-Riemannian manifolds. Key implications:

• The Myers-type finiteness theorem for shrinking solitons connects geometric flow theory to fundamental group constraints,
• The lack of explicit non-Riemannian soliton examples motivates further research into explicit geometry and the analytic structure of Ricci solitons outside Riemannian frameworks,
• The theory paves the way for studying singularity formation, canonical metrics, and long-term evolution under Finsler–Ricci flows, complementing advances in Riemannian and Lorentzian Ricci soliton theory.

The Finslerian perspective expands geometric flow theory by incorporating the greater generality and richness of Finsler structures, with implications for both geometric topology and metric geometry (Bidabad et al., 2018).