[Paper Review] A comparison theorem for Finsler submanifolds and its applications

Wei Zhao|arXiv (Cornell University)|Oct 29, 2017

Advanced Differential Geometry Research23 references3 citations

TL;DR

This paper establishes a Finsler version of the Heintze-Karcher volume comparison theorem using the conormal bundle of a submanifold, enabling lower bounds on closed geodesic lengths and injective radius in Finsler manifolds. The key contribution is a comparison estimate involving flag curvature, T-curvature, and volume forms, extending classical Riemannian results to non-Riemannian Finsler geometry with applications to global geometry and curvature bounds.

ABSTRACT

In this paper, we consider the conormal bundle over a submanifold in a Finsler manifold and establish a volume comparison theorem. As an application, we derive a lower estimate for length of closed geodesics in a Finsler manifold. In the reversible case, a lower bound of injective radius is also obtained.

Motivation & Objective

• To extend the Heintze-Karcher comparison theorem from Riemannian to Finsler geometry by analyzing the conormal bundle of submanifolds.
• To derive volume comparison estimates in Finsler manifolds using flag curvature and T-curvature.
• To establish lower bounds for the length of closed geodesics in closed Finsler manifolds.
• To obtain an injective radius estimate in the reversible Finsler case using the same comparison framework.
• To analyze the geometric behavior of the conormal bundle and its volume under curvature and uniform constant constraints.

Proposed method

• Define the conormal bundle $\mathcal{V}^*N$ as the Legendre transform image of the normal cone bundle $\mathcal{V}N$ in a Finsler manifold.
• Introduce the co-mean curvature $H_\xi$ along conormal directions $\xi \in \mathcal{V}^*N \setminus 0$ to quantify submanifold geometry.
• Use the T-curvature $\mathbf{T}$ to measure nonlinearity of the Chern connection and control curvature deviation in Finsler geometry.
• Derive a volume comparison inequality involving $\mathfrak{s}_\delta(t)$, the solution to $y'' + \delta y = 0$, and the co-mean curvature $H_{\xi_0}$.
• Apply the uniform constant $\Lambda_F$ to bound volume distortion between different metrics and control the conormal bundle volume.
• Utilize the Holmes-Thompson and Busemann-Hausdorff volume forms to generalize the comparison to non-Riemannian settings.

Experimental results

Research questions

• RQ1 Can a Heintze-Karcher-type volume comparison theorem be extended to Finsler manifolds using the conormal bundle?
• RQ2 How does the T-curvature influence the volume growth and geodesic length estimates in Finsler geometry?
• RQ3 What lower bounds on closed geodesic length can be derived from curvature and diameter constraints in Finsler manifolds?
• RQ4 How does the injective radius in a reversible Finsler manifold relate to curvature and volume?
• RQ5 What role does the uniform constant $\Lambda_F$ play in controlling volume distortion in the conormal bundle?

Key findings

• For a closed Finsler $m$-manifold with flag curvature $\mathbf{K} \geq \delta$, diameter $d$, and volume $\mu(M)$, the volume is bounded above by an integral involving $\mathfrak{s}_\delta(t)$ and the co-mean curvature $H_{\xi_0}$.
• When $k=0$, the volume bound reduces to $\mu(M) \leq \int_{S_xM} e^{-\tau(\dot{\gamma}_y(t))} d\nu_x(y) \int_0^d \mathfrak{s}_\delta^{m-1}(t) dt$, generalizing the Riemannian case.
• For $k \geq 1$, the volume satisfies $\mu(M) \leq c_{m-k-1} \cdot \Lambda_F^{(3m+k)/2} \cdot \bar{\mu}(N) \cdot \int_0^{\min\{d, \zeta(\xi_0)\}} \left(\mathfrak{s}'_\delta - \frac{H_{\xi_0}}{k}\mathfrak{s}_\delta\right)^k(t) \cdot \mathfrak{s}_\delta^{m-k-1}(t) dt$, incorporating curvature and mean curvature.
• A lower bound for the length of any closed geodesic $\gamma$ is derived as $L_g(\gamma) \geq \frac{(m-1)V}{c_{m-2} \mathfrak{s}_\delta^{m-1}(\min\{d, \pi/(2\sqrt{\delta})\})}$, extending the Riemannian result.
• In the reversible case, the injective radius is bounded below using the same comparison framework, avoiding Toponogov’s theorem.
• For Randers metrics $F = \alpha + \beta$, the conormal bundle volume is bounded by $c_{m-k-1} \cdot (1 - b(x))^{-(m-k+1)/2}$, where $b(x) = \|\beta\|_\alpha(x)$.