In mathematics, particularly differential geometry, a Finsler manifold is a differentiable manifold M with a Banach norm defined over each tangent space, smoothly depending on position, and (usually) assumed to satisfy the following condition:
- For each point x of M, and for every nonzero vector v in the tangent space TxM, the Hessian of the function L:TxM → R given by
- is positive definite at v. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. In Mathematics, Banach spaces (ˈbanax named after Polish Mathematician Stefan Banach) are one of the central objects of study in Functional analysis In Mathematics, the tangent space of a Manifold is a concept which facilitates the generalization of vectors from Affine spaces to general manifolds since In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Mathematics, the tangent space of a Manifold is a concept which facilitates the generalization of vectors from Affine spaces to general manifolds since In Mathematics, the Hessian matrix is the Square matrix of second-order Partial derivatives of a function. In mathematics positive definite may refer to Positive-definite matrix Positive-definite function Positive definite
The above condition implies that the norm function satisfies the triangle inequality. In Mathematics, the triangle inequality states that for any Triangle, the length of a given side must be less than or equal to the sum of the other two sides but greater The proof of this is not completely trivial.
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Examples
- Riemannian manifolds (but not pseudo-Riemannian manifolds) are special cases of Finsler manifolds. In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M In Differential geometry, a pseudo-Riemannian manifold (also called a semi-Riemannian manifold) is a generalization of a Riemannian manifold.
- Randers manifolds
Geodesics
The length of γ, a differentiable curve in M, is given by
Length is invariant under reparametrization. In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object Parameterization (or parametrization parameterisation in British English) is the process of defining or deciding the Parameters - usually of some model - that are Assuming the above condition on the Hessian, geodesics are locally length-minimizing curves with constant speed, or equivalently, curves whose energy function
is extremal (in the sense that its functional derivative vanishes). In Mathematics, a geodesic /ˌdʒiəˈdɛsɪk -ˈdisɪk/ -dee-sik is a generalization of the notion of a " straight line " to " curved spaces In Mathematics and theoretical Physics, the functional derivative is a generalization of the Directional derivative.
See also
- Metric tensor, used for differentiable manifolds with inner-product norms. In the mathematical field of Differential geometry, a metric tensor is a type of function defined on a Manifold (such as a Surface in space In Mathematics, an inner product space is a Vector space with the additional Structure of inner product.
External links
- Z. Shen's Finsler Geometry Website.
References
- D. Bao, S. S. Chern and Z. Shen, An Introduction to Riemann-Finsler Geometry, Springer-Verlag, 2000. ISBN 0-387-98948-X.
- S. Chern: Finsler geometry is just the Riemannian geometry without the quadratic restriction, Notices AMS, 43 (1996), pp. 959-63.
- H. Rund. The Differential Geometry of Finsler Spaces, Springer-Verlag, 1959. ASIN B0006AWABG.
- Z. Shen, Lectures on Finsler Geometry, World Scientific Publishers, 2001. ISBN 981-02-4531-9.
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