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Finsler Geometry: Simple Definition

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Finsler geometry, named after Paul Finsler’s 1918 doctoral thesis, has become a popular area of research in the last couple of decades. Many Finsler metrics occur naturally in convexity theory and topology; This geometry can also be seen in various other areas of mathematics, including discrete group representation and dynamical systems, as well as practical applications in physics and other natural sciences.

The idea behind Finsler geometry is quite simple: it is the geometry of a simple integral. However, the calculations are often extremely complicated, revolving around tensors. This often discourages beginners (Shen, 2001), but anyone with a background in differential geometry and tensor calculus should be able to grasp the basics.

Riemannian Geometry vs. Finsler Geometry

Riemannian geometry is the study of smooth manifolds with a Riemannian metric g, an inner product on each tangent space which varies smoothly over the entire manifold. The length of a curve λ ↦ γ(λ) is independent of the curve’s parametrization and can be defined as
finsler geometry

The above statement can be generalized, allowing the metric tensor’s components to vary direction as well as position. This is the reason that Finsler geometry is sometimes called a generalization of Riemannian geometry

Roguer Brussee posted a really good description, which sums up the intuitive differences between the two types of geometry:

“I live in a small city in the east of Holland, a rather small country: the geographic distance from east to west, is about 200km. Geographic distance is what Riemannian distance is modeling and, of course the distance from west to east of the country is the same distance, i.e. about 200 km.
Many of my friends, live in the west, as indeed most of the people in Holland do. I and most of my friends in the east regularly drive to our friends in the west. Because we easterners are used to regular drives to the west, the perception is that it is not so far. My friends in the west, however, are used to most people living near in the west and seldom drive to the east. In their perception, it is actually quite far away. This shows that the _perception_ of distance is more like a Finsler geometry:
d_{percept}(east, west) < d_{percept}(west, east)."

References

Brussee, Rogier. (2013). Re: What is the basic difference between Riemann and Finsler geometry?. Retrieved from: https://www.researchgate.net/post/What_is_the_basic_difference_between_Riemann_and_Finsler_geometry2/521e782ed11b8b6a2599149c/citation/download.
Chern, S. (1996). Finsler Geometry Is Just Riemannian Geometry without the Quadratic Restriction. Retrieved February 14, 2021 from: http://www.ams.org/notices/199609/chern.pdf
Gudapati, N. (2018). Gravitational Lensing of Stationary Spacetimes: A Finsler Geometry Approach. BHI Colloquium. Retrieved February 14, 2021 from: https://scholar.harvard.edu/files/nishanth/files/gudapati_bhi.pdf
Shen, Z. (2001). Lectures on Finsler Geometry. World Scientific.

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Stephanie Glen. "Finsler Geometry: Simple Definition" From GeometryHowTo.com: Geometry for the rest of us! https://www.geometryhowto.com/finsler-geometry-simple-definition/
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