Finsler metrics

A Finsler metric assigns a smooth Minkowski norm to every tangent space. It keeps the positivity, homogeneity and convexity needed for geometry, but the norm need not come from an inner product and need not assign equal cost to opposite directions.

What Riemann set aside

Riemann introduced the $n$ -dimensional case in his Habilitationsvortrag of 1854, taking the line element to be the square root of a positive definite quadratic form in the coordinate differentials, which is the definition still in use today. The more general possibility was not lost on him. A line element that is, for instance, the fourth root of a quartic form appears in the same lecture, and he set it aside there on the grounds that it would demand considerable work while yielding comparatively little new insight into space.

The usual reference point for the general definition is Finsler’s 1918 thesis, written under Carathéodory, which arrived at these metrics from the calculus of variations rather than from a wish to generalise for its own sake — in that setting an integrand depending on direction as well as position is the ordinary case, not the exotic one. Chern later put the relationship between the two theories into a title: Finsler geometry is just Riemannian geometry without the quadratic restriction (Notices of the AMS, 1996).

Cost that depends on the direction

A Riemannian norm comes from an inner product. It may vary with position and heading, but it assigns an infinitesimal step and its reverse the same length; consequently its distance is symmetric. This is the right model for length, but not for every quantity accumulated along a path.

It stops describing the situation as soon as the quantity of interest is an effort or a time rather than a length. A hiker crossing a mountainside spends more of both climbing a slope than descending the same slope, so the cost of a step up and the cost of the step back down are simply different numbers, and no symmetric metric can hold both of them at once. Matsumoto (1989) made the observation precise, showing that if time is measured on a hillside in place of distance, the slope becomes a Finsler surface. Replacing the length of a tangent vector with the effort of travelling along it removes the difficulty at its source, since effort has no particular reason to be symmetric in the first place. The definition below keeps enough regularity and convexity for geodesics, parallel transport and curvature to remain available.

The defining conditions

A Finsler metric is a non-negative function $F(x,v)$ on the tangent bundle, positive for $v \neq 0$ and smooth away from the zero section. Its first structural condition is positive homogeneity in the velocity,

F(x,\lambda v) = \lambda\,F(x,v), \qquad \lambda > 0,

which is what makes the cost of a path, $\int F(\gamma,\dot\gamma)\,dt$ , independent of every orientation-preserving reparametrisation. Positive homogeneity says nothing about $F(x,-v)$ ; equality of $F(x,v)$ and $F(x,-v)$ is the additional reversibility condition that a general Finsler metric is not required to satisfy.

The second is strong convexity, meaning that the fundamental tensor

g_{ij}(x,v) = \frac{\partial^2 E}{\partial v^i \partial v^j}(x,v) = \tfrac{1}{2}\,\frac{\partial^2 F^2}{\partial v^i \partial v^j}(x,v)

is positive definite for every $v \neq 0$ . This strong-convexity condition makes the unit sphere curved outward in every tangent direction and makes the velocity Hessian of the energy invertible, so that the Euler–Lagrange equations can be solved for the acceleration and the geodesic equation is a genuine second-order system instead of a degenerate one. It also produces $g$ itself, which plays the role of the Riemannian metric tensor in every formula further down the subject, with the single difference that it depends on the direction $v$ as well as the point $x$ .

Riemannian geometry is the special case in which $g$ is independent of the reference direction $v$ .

The indicatrix

Because $F$ is homogeneous, it is determined completely by its unit level set, and that set is the most useful picture of the metric available. The indicatrix at $x$ is $\mathcal{I}_x = \{\, y \in T_xM : F_x(y) = 1 \,\}$ , the set of directions we can afford at one unit of cost, and scaling recovers the cost of every other vector from it. Strong convexity implies that this set bounds a smooth, strictly convex body containing the origin. The converse needs care: ordinary strict convexity alone does not guarantee positive-definite second fundamental form everywhere.

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finsler-indicatrix-field

Drag the point. The left panel is the domain, with the indicatrix drawn at a grid of its points — the definition itself, one norm per tangent space, varying smoothly with position — and the metric at the point you are on is written out underneath. Moving right raises

a

, the three-fold lobe, which is a shape no ellipse has at any eccentricity, so the curve is visibly not from a quadratic form. Moving up raises

b

, which pushes the origin off centre and is the asymmetry. Both vanish at the bottom-left corner, where the indicatrix is the Euclidean unit circle drawn faint beside it.

For a Riemannian metric the indicatrix is an ellipsoid centred on the origin, and it is precisely that centring which expresses the symmetry, since every direction then has its opposite at the same distance. Asymmetry is the failure of that central symmetry — the indicatrix is no longer carried onto itself by $y \mapsto -y$ — and it can fail in complicated ways, since a convex body need not have a centre at all. For the Randers family on the next page it fails in the simplest way available: the same ellipsoid translated bodily off the origin, and the translation is the drift.

How far the deformation can be taken is itself decided by strong convexity, and the bound is tighter than it looks. For an indicatrix written as a polar curve $r(\theta)$ the condition reads $r^2 + 2r'^2 - r\,r'' > 0$ , which for a three-fold lobe $r = 1 - a\cos 3\theta$ is smallest at $\cos 3\theta=1$ . Its minimum is $1 - 11a + 10a^2$ , so the range used here admits a metric only for $a < 1/10$ . The figure above runs to $a = 0.06$ , and the generator behind it assembles $g$ from the library at every sampled point and direction and refuses to write the asset if the smallest eigenvalue is not positive.

The origin has to remain strictly inside the body bounded by the indicatrix. If the boundary reaches it, a non-zero unit-control velocity produces zero ground velocity; the navigation norm is no longer a smooth, strongly convex Finsler metric. This becomes the drift bound on the next page.

Let $M$ be an $n$ -dimensional manifold, $TM$ its tangent bundle. A Finsler metric is a function $F\colon TM \to [0,\infty)$ , smooth on $TM \setminus \{0\}$ , positively homogeneous of degree one in the velocity, and such that $g_{ij} = \tfrac{1}{2}\,\partial^2 F^2/\partial v^i \partial v^j$ is positive definite for all $v \neq 0$ . Equivalently, each tangent space carries a Minkowski norm and $F$ is the smooth assignment of one to every point.
Euler’s theorem on homogeneous functions gives two identities used throughout. Since $F^2$ is homogeneous of degree two, $g_{ij}$ is homogeneous of degree zero — it depends on the direction of $v$ but not on its length — and
$g_{ij}(x,v)\,v^i v^j = F^2(x,v).$
Riemannian geometry is the case $F(x,v) = \sqrt{g_x(v,v)}$ with $g$ independent of $v$ . Between it and the general case sit the Berwald metrics, whose spray is quadratic in $v$ although $F$ is not; they are where several Riemannian statements survive intact, and they recur on the transport page for that reason.

In the library

FinslerMetric is the base class, and it requires exactly one method, metric_fn(x, v), returning $F$ . Everything above is then taken from that by differentiation: energy squares and halves it, inner_product is the velocity Hessian of the energy, and spray follows from those. A metric with no structure whatever is therefore the base case rather than a special one, and the named families — Randers, Riemannian — exist because a closed form is cheaper to evaluate and easier to constrain, not because anything downstream needs them to exist.

Two numerical qualifications are visible in the interface. First, implementations ofmetric_fn must define a gradient-safe value at $v=0$ , even though the mathematical derivatives of a Finsler norm are required only away from the zero section. The concrete metrics use guarded norms and return zero in the forward pass; an arbitrary subclass is responsible for the same contract. Second, the linear system behind the spray carries a Tikhonov term scaled by $\operatorname{tr} H / D$ instead of a fixed constant, so that the relative size of the perturbation is similar for metrics of different magnitude, with the scale floored at one. The resulting spray is a regularised numerical approximation, and curvature magnifies its bias near a degenerate direction.