Every Finsler manifold becomes an intrinsic quasimetric space when the distance between two points is defined as the infimum length of the curves that join them. The Finsler metric F is also required to be smooth , more precisely:.
Tamássy : Finsler geometry in the tangent bundle
Here the Hessian of F 2 at v is the symmetric bilinear form. If F is strongly convex, then it is a Minkowski norm on each tangent space. A reversible Finsler metric defines a norm in the usual sense on each tangent space. Let M , d be a quasimetric so that M is also a differentiable manifold and d is compatible with the differential structure of M in the following sense:.
The Finsler function F obtained in this way restricts to an asymmetric typically non-Minkowski norm on each tangent space of M.
An Introduction to Riemann-Finsler Geometry
In analogy with the Riemannian case, there is a version. From Wikipedia, the free encyclopedia. For a smooth function u on M , the gradient vector and the Finsler-Laplacian of u is defined by. In particular, on M u we have.
An introduction to Finsler geometry / Xiaohuan Mo.
Let M , F be a Finsler manifold. Define the distance function by. Lemma 2.
Then on M u we have. Theorem 3. Using 2. We refer to [ 14 ] for details. This yields. Combining 3. Proposition 3. Proposition 4. For two positive integrable functions f and g , if f g is monotone increasing, then the function. Theorem 4.
Since the Stokes formula still holds for Lipschitz continuous functions, we have. Combining this and 4.
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For example,. Moreover, the corresponding Laplacian comparison theorem still holds [ 6 ]. So the proof is omitted here. Corollary 4. It is well known that.
Remark 4. In [ 15 ], Milnor proved that the fundamental group of a compact Riemannian manifold of negative sectional curvature has exponential growth.
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Then this result was generalized to the case of negative Ricci curvature and nonpositive sectional curvature in [ 16 ] and [ 17 ]. The key point of the proof is to give a lower bound estimate for the volume of the geodesic balls of the universal covering space. In [ 3 ] and [ 4 ], the results were also generalized to the Finsler setting.
Suppose that one of the following two conditions holds :. Theorem 5. Integrating both sides of 5. Pure Appl. Wu BY: Volume form and its applications in Finsler geometry. Ohta S: Finsler interpolation inequalities. Partial Differ. Ding Q: A new Laplacian comparison theorem and the estimate of eigenvalues. B , Abstract No. Yau ST: Some function-theoretic properties of complete Riemannian manifold and their applications to geometry. Indiana Univ.
Xing H: The geometric meaning of Randers metrics with isotropic S -curvature. Acta Math. Milnor J: A note on curvature and fundamental group. Yang YH: On the growth of the fundamental groups on nonpositive curvature manifolds.
Related An Introduction to Finsler Geometry
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