Finsler Space

A general space based on the Line Element

$$ds=F(x^1, \ldots, x^n; dx^1, \ldots, dx^n),$$

with $F(x,y)>0$ for $y\not=0$ a function on the Tangent Bundle $T(M)$, and homogeneous of degree 1 in $y$. Formally, a Finsler space is a Differentiable Manifold possessing a Finsler Metric. Finsler geometry is Riemannian Geometry without the restriction that the Line Element be quadratic and of the form

$$F^2=g_{ij}(x)\,dx^i\,dx^j.$$

A compact boundaryless Finsler space is locally Minkowskian Iff it has 0 ``flag curvature.''

See also Finsler Metric, Hodge's Theorem, Riemannian Geometry, Tangent Bundle

References

Akbar-Zadeh, H. ``Sur les espaces de Finsler à courbures sectionnelles constantes.'' Acad. Roy. Belg. Bull. Cl. Sci. 74, 281-322, 1988.

Bao, D.; Chern, S.-S.; and Shen, Z. (Eds.). Finsler Geometry. Providence, RI: Amer. Math. Soc., 1996.

Chern, S.-S. ``Finsler Geometry is Just Riemannian Geometry without the Quadratic Restriction.'' Not. Amer. Math. Soc. 43, 959-963, 1996.

Iyanaga, S. and Kawada, Y. (Eds.). ``Finsler Spaces.'' §161 in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 540-542, 1980.