Finsler manifold

Examples

Riemannian manifolds are special cases of Finsler manifolds, since their unit balls are hyperellipsoids that smoothly vary over space. More generally, smooth submanifolds (including open subsets) of a normed vector space of finite dimension are Finsler manifolds if the norm of the vector space is smooth outside the origin.

Pseudo-Riemannian manifolds are not Finsler, since their unit balls are hyperboloids, which are not bounded.

Randers manifolds

Let ${\displaystyle (M,\alpha )}$ be a Riemannian manifold and ${\displaystyle \beta }$ a differential one-form on M such that $${\displaystyle \mathrm{\forall }v\in T_{x}M\setminus \{0\},\alpha (x)(v,v)>\beta (x)(v{)}^2}$$then

${\displaystyle F(x,v):=\sqrt{\alpha (x)(v,v)}+\beta (x)(v)}$

defines a Randers metric, and ${\displaystyle (M,F)}$ is a Randers manifold. They are not reversible when ${\displaystyle \beta \ne 0}$. Unit balls of a Randers manifold are off-center ellipsoids.

Zermelo's navigation problem asks for the fastest path for a boat to go from point A to B on a flat ocean, when there is an ocean current velocity field. If the velocity field is smaller than the boat's maximum speed, then the geometry of the problem is a Randers manifold.

Kropina space

Let ${\displaystyle (M,\alpha )}$ be a Riemannian manifold and ${\displaystyle \beta }$ a differential one-form on M such that $${\displaystyle \mathrm{\forall }v\in T_{x}M\setminus \{0\},\alpha (x)(v,v)>\beta (x)(v{)}^2}$$then

${\displaystyle F(x,v):=\alpha (x)(v,v)/\beta (x)(v)}$

defines a Kropina space. Strictly speaking, it is not a Finsler geometry, because the unit balls are ellipsoids that are so off-center that the origins fall on their boundaries, meaning that ${\displaystyle F(x,v)}$ is only defined in an open half-space of ${\displaystyle T_{x}M}$. However, many concepts of Finsler geometry still apply to Kropina spaces. In Zermelo's navigation problem, this corresponds to the case where the ocean current is everywhere as fast as the boat's maximum speed.

(α, β)-Finsler manifolds

Let ${\displaystyle (M,\alpha )}$ be a Riemannian manifold and ${\displaystyle \beta }$ be a differential one-form on ${\displaystyle M}$ such that for all ${\displaystyle x\in M}$:

${\displaystyle \mathrm{\forall }v\in T_{x}M\setminus \{0\},\alpha (x)(v,v)>\beta (x)(v{)}^2.}$

If ${\displaystyle \varphi }$ is a smooth positive function defined on an open interval containing ${\displaystyle [-1,1]}$, then the function ${\displaystyle F:TM\to [0,\mathrm{\infty })}$ defined by:

${\displaystyle F(x,v)=\alpha (x)(v,v)\varphi \left(\frac{\beta (x)(v)}{\alpha (x)(v,v)}\right)}$

defines an (α, β)-metric. Both Randers (${\displaystyle \varphi (s)=1+s}$ ) and Kropina (${\displaystyle \varphi (s)=1/s}$) spaces are special cases of (α, β)-metrics.

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

  ${\displaystyle \frac{1}{C}\Vert \varphi (y)-\varphi (x)\Vert \le d(x,y)\le C\Vert \varphi (y)-\varphi (x)\Vert .}$

• 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

${\displaystyle F(x,v):=\underset{t\to 0+}{lim}\frac{d(\gamma (0),\gamma (t))}{t},}$

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

${\displaystyle d_L(x,y):=inf\left\{{\int }_0^{1}F\left(\gamma (t),\dot{\gamma }(t)\right)dt|\gamma \in C^1([0,1],M),\gamma (0)=x,\gamma (1)=y\right\},}$

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

Structures

Many structures in Riemannian geometry have counterparts in Finsler geometry.

Duality

Convex duality (or Legendre transformation) creates a bijection ${\displaystyle T_{x}M\leftrightarrow T_x^{\ast }M}$. For any unit-length ${\displaystyle v\in T_{x}M}$, construct the plane that is tangent to the unit ball at ${\displaystyle v}$. That is the convex dual ${\displaystyle v^{\ast }}$. Now extend this bijection by 1-homogeneity to a full bijection ${\displaystyle T_{x}M\leftrightarrow T_x^{\ast }M}$.

The convex dual of the unit ball is also strongly convex and contains the origin, thus defining a dual Finsler (or co-Finsler) metric ${\displaystyle H}$ on the cotangent bundle ${\displaystyle T^{\ast }M}$. Over a given manifold, the convex dual operation creates a bijection between the Finsler metrics and dual Finsler metrics that can be defined over the manifold.

Flatness

A Finsler space is locally projectively flat (aka locally projectively Euclidean, with rectilinear extremals, or with straight geodesics) iff there exists an atlas ${\displaystyle \{U_i,{\varphi }_i\}}$, such that any geodesic in a coordinate chart ${\displaystyle \gamma :I\to U_i}$ is, in coordinates, of the form ${\displaystyle t\mapsto f(t)\overrightarrow{a}+\overrightarrow{b}}$ for some ${\displaystyle \overrightarrow{a},\overrightarrow{b}\in {\mathbb{R}}^n}$ and some strictly monotonic ${\displaystyle f:I\to \mathbb{R}}$. It is globally projectively flat iff there exists a single coordinate system that makes it locally projectively flat around every point.

The question of flatness was first asked in the context of Hilbert's fourth problem.

Geodesics

Due to the homogeneity of F, the arc length

${\displaystyle L[\gamma ]:={\int }_a^{b}F\left(\gamma (t),\dot{\gamma }(t)\right)dt}$

of a differentiable curve γ: [a, b] → M in M is invariant under strictly positively monotonic reparametrizations. In particular, it allows the unit-speed (or arc length) parameterization. Note that since a Finsler manifold might not be reversible, the arc length is only well-defined for a directed differentiable curve.

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

${\displaystyle E[\gamma ]:=\frac{1}{2}{\int }_a^b{F}^2\left(\gamma (t),\dot{\gamma }(t)\right)dt}$

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. This can be interpreted as a stationary action principle.

Geodesic flow

As in the case of Riemannian geometry, at any point in any direction, there exists a unique geodesic going that way. Using unit-speed parametrization, it creates a transport of the unit-length tangent vector, which extends by 1-homogeneity to a transport of the tangent space. This is the geodesic flow on the tangent bundle. Taking the convex dual, it defines a cogeodesic flow on the cotangent bundle.

A version of the Huygens–Fresnel principle applies to the (co)geodesic flow. Specifically, the Finsler metric induces a contact 1-form on the space of unit-length tangent vectors, which by duality is a contact 1-form on the space of unit-length cotangent vectors. Then the (co)geodesic flow is the corresponding Reeb vector flow.

Geodesic spray

The geodesic flow is a flow on the tangent bundle ${\displaystyle TM}$. Equivalently, it is the flow of a section of the double tangent bundle ${\displaystyle TTM}$. This section of ${\displaystyle TTM}$ is the geodesic spray.

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

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

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

${\displaystyle g_{ij}(x,v):=g_v\left({\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^i}|}_x,{\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^j}|}_x\right).}$

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

${\displaystyle {H|}_{(x,v)}:={v^i\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^i}|}_{(x,v)}-{2G^i(x,v)\frac{\mathrm{\partial }}{\mathrm{\partial }{v}^i}|}_{(x,v)},}$

where the local spray coefficients Gⁱ are given by

${\displaystyle G^i(x,v):=\frac{1}{4}{g}^{ij}(x,v)\left(2\frac{\mathrm{\partial }{g}_{jk}}{\mathrm{\partial }{x}^{\ell }}(x,v)-\frac{\mathrm{\partial }{g}_{k\ell }}{\mathrm{\partial }{x}^j}(x,v)\right)v^k{v}^{\ell }.}$

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

${\displaystyle v:T(\mathrm{T}M\setminus \{0\})\to T(\mathrm{T}M\setminus \{0\});v:=\frac{1}{2}(I+{\mathcal{L}}_{H}J).}$

In analogy with the Riemannian case, there is a version

${\displaystyle D_{\dot{\gamma }}{D}_{\dot{\gamma }}X(t)+R_{\dot{\gamma }}\left(\dot{\gamma }(t),X(t)\right)=0}$

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 the Hopf–Rinow theorem, the manifold can be covered with open sets such that any two points in a set is connected by a unique length minimizing curve. 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.

Connection

The geodesic flow determines how a path transports its own velocity vector. It does not determine how it transports other vectors. A Finsler connection extends it to a transport, by defining an Ehresmann connection on the double tangent bundle. This is not necessarily an affine connection, so the transport may be nonlinear.

Consequently, the covariant derivative may also be nonlinear. The curvature form is also defined.