Finsler manifold

In mathematics, particularly differential geometry, a Finsler manifold is a differentiable manifold together with the structure of an intrinsic quasimetric space in which the length of any rectifiable curve γ : [a,b] → M is given by the length functional

L[\gamma] = \int_a^b F(\gamma(t),\dot{\gamma}(t))\,dt,

where F(x, · ) is a Minkowski norm (or at least an asymmetric norm) on each tangent space T_xM. Finsler manifolds non-trivially generalize Riemannian manifolds in the sense that they are not necessarily infinitesimally Euclidean. This means that the (asymmetric) norm on each tangent space is not necessarily induced by an inner product (metric tensor).

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

Definition

A Finsler manifold is a differentiable manifold M together with a Finsler function F defined on the tangent bundle of M so that for all tangent vectors v,

F is smooth on the complement of the zero section of TM.
F(v) ≥ 0 with equality if and only if v = 0 (positive definiteness).
F(λv) = λF(v) for all λ ≥ 0 (but not necessarily for λ < 0) (homogeneity).
F(v + w) ≤ F(v) + F(w) for all w at the same tangent space with v (subadditivity).

In other words, F is an asymmetric norm on each tangent space. Typically one replaces the subadditivity with the following strong convexity condition:

For each tangent vector v, the hessian of F² at v is positive definite.

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

\mathbf{g}_v(X,Y) := \frac{1}{2}\left.\frac{\partial^2}{\partial s\partial t}\left[F(v + sX + tY)^2\right]\right|_{s=t=0},

also known as the fundamental tensor of F at v. Strong convexity of F² implies the subadditivity with a strict inequality if u/F(u) ≠ v/F(v). If F² is strongly convex, then F 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

Normed vector spaces of finite dimension, such as Euclidean spaces, whose norms are smooth outside the origin.
Riemannian manifolds (but not pseudo-Riemannian manifolds) are special cases of Finsler manifolds.

Randers manifolds

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

\|b\|_a := \sqrt{a^{ij}b_i b_j} < 1,

where $(a^{ij})$ is the inverse matrix of $(a_{ij})$ and the Einstein notation is used. Then

F(x,v) := \sqrt{a_{ij}(x)v^iv^j} + b_i(x)v^i

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

Smooth quasimetric spaces

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 x,y ∈ U

\frac{1}{C}\|\varphi(y)-\varphi(x)\| \leq d(x,y) \leq C\|\varphi(y)-\varphi(x)\|.

The function d : M × M →[0,∞] is smooth in some punctured neighborhood of the diagonal.

Then one can define a Finsler function F : TM →[0,∞] by

F(x,v) := \lim_{t\to 0+} \frac{d(\gamma(0),\gamma(t))}{t},

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 d_L: M × M → [0, ∞] of the original quasimetric can be recovered from

d_L(x,y) := \inf\left\{\ \left.\int_0^1 F(\gamma(t),\dot\gamma(t)) \, dt \ \right| \ \gamma\in C^1([0,1],M) \ , \ \gamma(0)=x \ , \ \gamma(1)=y \ \right\},

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

Geodesics

Due to the homogeneity of F the length

L[\gamma]:=\int_a^b F(\gamma(t),\dot{\gamma}(t))\, dt

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

E[\gamma]:=\frac{1}{2}\int_a^b F^2(\gamma(t),\dot{\gamma}(t))\, dt

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

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

g_{ik}\Big(\gamma(t),\dot\gamma(t)\Big)\ddot\gamma^i(t) + \left(\frac{\partial g_{ik}}{\partial x^j}\Big(\gamma(t),\dot\gamma(t)\Big) - \frac{1}{2}\frac{\partial g_{ij}}{\partial x^k}\Big(\gamma(t),\dot\gamma(t)\Big) \right)\dot\gamma^i(t)\dot\gamma^j(t) = 0,

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

g_{ij}(x,v) := g_v\left(\tfrac{\partial}{\partial x^i}\big|_x,\tfrac{\partial}{\partial x^j}\big|_x\right).

Assuming the strong convexity of F²(x,v) with respect to v∈T_xM, the matrix g_ij(x,v) is invertible and its inverse is denoted by g^ij(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

H|_{(x,v)} := v^i\tfrac{\partial}{\partial x^i}\big|_{(x,v)} - \ 2G^i(x,v)\tfrac{\partial}{\partial v^i}\big|_{(x,v)},

where the local spray coefficients Gⁱ are given by

G^i(x,v) := \frac{g^{ij}(x,v)}{4}\left(2\frac{\partial g_{jk}}{\partial x^\ell}(x,v) - \frac{\partial g_{k\ell}}{\partial x^j}(x,v)\right)v^k v^\ell.

The vector field H on TM/0 satisfies JH = V and [V,H] = 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

v:T(TM\setminus 0)\to T(TM\setminus 0) \quad ; \quad v := \tfrac{1}{2}\big( I + \mathcal L_H J \big).

In analogy with the Riemannian case, there is a version

D_{\dot\gamma}D_{\dot\gamma}X(t) + R_{\dot\gamma}(\dot\gamma(t),X(t)) = 0

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

By Hopf–Rinow theorem there always exist length minimizing curves (at least in small enough neighborhoods) on (M, F). 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 (x, v) ∈ 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

↑ 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.

References

Antonelli, P. L., ed. (2003), Handbook of Finsler geometry. Vol. 1, 2, Boston: Kluwer Academic Publishers, ISBN 978-1-4020-1557-1, MR 2067663
D. Bao, S. S. Chern and Z. Shen, An Introduction to Riemann–Finsler Geometry, Springer-Verlag, 2000. ISBN 0-387-98948-X.
Cartan, Elie (1933), "Sur les espaces de Finsler", C. R. Acad. Sci., Paris 196: 582–586, Zbl 0006.22501
S. Chern: Finsler geometry is just Riemannian geometry without the quadratic restriction, Notices AMS, 43 (1996), pp. 959–63.
Finsler, Paul (1918), Über Kurven und Flächen in allgemeinen Räumen, Dissertation, Göttingen, JFM 46.1131.02 (Reprinted by Birkhäuser (1951))
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.

External links

Hazewinkel, Michiel, ed. (2001), "Finsler space, generalized", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Z. Shen's Finsler Geometry Website.
The (New) Finsler Newsletter

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