Finsler manifold


In mathematics, particularly differential geometry, a Finsler manifold is a differentiable manifold M where a (possibly asymmetric) Minkowski norm F(x, −) is provided on each tangent space TxM, that enables one to define the length of any smooth curve γ : [a, b] → M as

Finsler manifolds are more general than Riemannian manifolds since the tangent norms need not be induced by inner products.

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.

Élie Cartan (1933) named Finsler manifolds after Paul Finsler, who studied this geometry in his dissertation (Finsler 1918).

Definition edit

A Finsler manifold is a differentiable manifold M together with a Finsler metric, which is a continuous nonnegative function F: TM → [0, +∞) defined on the tangent bundle so that for each point x of M,

In other words, F(x, −) is an asymmetric norm on each tangent space TxM. The Finsler metric F is also required to be smooth, more precisely:

  • F is smooth on the complement of the zero section of TM.

The subadditivity axiom may then be replaced by the following strong convexity condition:

Here the Hessian of F2 at v is the symmetric bilinear form


also known as the fundamental tensor of F at v. Strong convexity of F implies the subadditivity with a strict inequality if uF(u)vF(v). If F is strongly convex, then it is a Minkowski norm on each tangent space.

A Finsler metric is reversible if, in addition,

  • F(−v) = F(v) for all tangent vectors v.

A reversible Finsler metric defines a norm (in the usual sense) on each tangent space.

Examples edit

Randers manifolds edit

Let   be a Riemannian manifold and b a differential one-form on M with


where   is the inverse matrix of   and the Einstein notation is used. Then


defines a Randers metric on M and   is a Randers manifold, a special case of a non-reversible Finsler manifold.[1]

Smooth quasimetric spaces edit

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:

  • Around any point z on M there exists a smooth chart (U, φ) of M and a constant C ≥ 1 such that for every xy ∈ U
  • The function dM × M → [0, ∞] is smooth in some punctured neighborhood of the diagonal.

Then one can define a Finsler function FTM →[0, ∞] by


where γ is any curve in M with γ(0) = x and γ'(0) = v. The Finsler function F obtained in this way restricts to an asymmetric (typically non-Minkowski) norm on each tangent space of M. The induced intrinsic metric dL: M × M → [0, ∞] of the original quasimetric can be recovered from


and in fact any Finsler function F: TM → [0, ∞) defines an intrinsic quasimetric dL on M by this formula.

Geodesics edit

Due to the homogeneity of F the length


of a differentiable curve γ: [a, b] → M in M is invariant under positively oriented reparametrizations. A constant speed curve γ is a geodesic of a Finsler manifold if its short enough segments γ|[c,d] are length-minimizing in M from γ(c) to γ(d). Equivalently, γ is a geodesic if it is stationary for the energy functional


in the sense that its functional derivative vanishes among differentiable curves γ: [a, b] → M with fixed endpoints γ(a) = x and γ(b) = y.

Canonical spray structure on a Finsler manifold edit

The Euler–Lagrange equation for the energy functional E[γ] reads in the local coordinates (x1, ..., xn, v1, ..., vn) of TM as


where k = 1, ..., n and gij is the coordinate representation of the fundamental tensor, defined as


Assuming the strong convexity of F2(x, v) with respect to v ∈ TxM, the matrix gij(x, v) is invertible and its inverse is denoted by gij(x, v). Then γ: [a, b] → M is a geodesic of (M, F) if and only if its tangent curve γ': [a, b] → TM∖{0} is an integral curve of the smooth vector field H on TM∖{0} locally defined by


where the local spray coefficients Gi are given by


The vector field H on TM∖{0} satisfies JH = V and [VH] = H, where J and V are the canonical endomorphism and the canonical vector field on TM∖{0}. Hence, by definition, H is a spray on M. The spray H defines a nonlinear connection on the fibre bundle TM∖{0} → M through the vertical projection


In analogy with the Riemannian case, there is a version


of the Jacobi equation for a general spray structure (M, H) in terms of the Ehresmann curvature and nonlinear covariant derivative.

Uniqueness and minimizing properties of geodesics edit

By Hopf–Rinow theorem there always exist length minimizing curves (at least in small enough neighborhoods) on (MF). Length minimizing curves can always be positively reparametrized to be geodesics, and any geodesic must satisfy the Euler–Lagrange equation for E[γ]. Assuming the strong convexity of F2 there exists a unique maximal geodesic γ with γ(0) = x and γ'(0) = v for any (xv) ∈ TM∖{0} by the uniqueness of integral curves.

If F2 is strongly convex, geodesics γ: [0, b] → M are length-minimizing among nearby curves until the first point γ(s) conjugate to γ(0) along γ, and for t > s there always exist shorter curves from γ(0) to γ(t) near γ, as in the Riemannian case.

Notes edit

  1. ^ Randers, G. (1941). "On an Asymmetrical Metric in the Four-Space of General Relativity". Phys. Rev. 59 (2): 195–199. doi:10.1103/PhysRev.59.195. hdl:10338.dmlcz/134230.

See also edit

  • Banach manifold – Manifold modeled on Banach spaces
  • Fréchet manifold – topological space modeled on a Fréchet space in much the same way as a manifold is modeled on a Euclidean space
  • Global analysis – which uses Hilbert manifolds and other kinds of infinite-dimensional manifolds
  • Hilbert manifold – Manifold modelled on Hilbert spaces

References edit

  • Antonelli, Peter L., ed. (2003), Handbook of Finsler geometry. Vol. 1, 2, Boston: Kluwer Academic Publishers, ISBN 978-1-4020-1557-1, MR 2067663
  • Bao, David; Chern, Shiing-Shen; Shen, Zhongmin (2000). An introduction to Riemann–Finsler geometry. Graduate Texts in Mathematics. Vol. 200. New York: Springer-Verlag. doi:10.1007/978-1-4612-1268-3. ISBN 0-387-98948-X. MR 1747675.
  • Cartan, Élie (1933), "Sur les espaces de Finsler", C. R. Acad. Sci. Paris, 196: 582–586, Zbl 0006.22501
  • Chern, Shiing-Shen (1996), "Finsler geometry is just Riemannian geometry without the quadratic restriction" (PDF), Notices of the American Mathematical Society, 43 (9): 959–63, MR 1400859
  • Finsler, Paul (1918), Über Kurven und Flächen in allgemeinen Räumen, Dissertation, Göttingen, JFM 46.1131.02 (Reprinted by Birkhäuser (1951))
  • Rund, Hanno (1959). The differential geometry of Finsler spaces. Die Grundlehren der Mathematischen Wissenschaften. Vol. 101. Berlin–Göttingen–Heidelberg: Springer-Verlag. doi:10.1007/978-3-642-51610-8. ISBN 978-3-642-51612-2. MR 0105726.
  • Shen, Zhongmin (2001). Lectures on Finsler geometry. Singapore: World Scientific. doi:10.1142/4619. ISBN 981-02-4531-9. MR 1845637.

External links edit