---
tag: 5R6QZ388BB
kind: concept
name: Finsler manifold
msc: [53B40, 53C60]
sources:
  - url: https://en.wikipedia.org/wiki/Finsler_manifold
same_as:
  wikidata: Q2498782
  wikipedia: Finsler manifold
---

## Summary

A Finsler manifold is a [[smooth manifold|F45VQ47GPV]] with a smoothly varying, strongly convex [[norm|0V2WP9B1MF]]-like function on each tangent space.
It generalises a [[Riemannian manifold|7CBT4SM83Y]], where the length of a tangent vector comes from an inner product.
Lengths of curves, distances and geodesics are defined as in Riemannian geometry.
But the unit ball in each tangent space may be any smooth strongly convex body, not only an ellipsoid.

## Explanations

### L1 human

In Riemannian geometry, the set of tangent vectors of length one at a point is always an ellipse or an ellipsoid.
Finsler geometry allows this unit set to be any smooth convex shape around the origin.
Then the cost of moving can depend on the direction, as when walking with or against a wind.
The length need not even be symmetric: going one way along a path can cost less than coming back.