Finsler manifold
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In
Finsler manifolds are more general than
Every Finsler manifold becomes an
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 metric, which is a continuous nonnegative function F: TM → [0, +∞) defined on the tangent bundle so that for each point x of M,
- F(v + w) ≤ F(v) + F(w) for every two vectors v,w tangent to M at x (subadditivity).
- F(λv) = λF(v) for all λ ≥ 0 (but not necessarily for λ < 0) (positive homogeneity).
- F(v) > 0 unless v = 0 (positive definiteness).
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 smoothon the complement of the zero section of TM.
The subadditivity axiom may then be replaced by the following strong convexity condition:
- For each tangent vector v ≠ 0, the positive definite.
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 u⁄F(u) ≠ v⁄F(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
- 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.
- Riemannian manifolds (but not pseudo-Riemannian manifolds) are special cases of Finsler manifolds.
Randers manifolds
Let be a Riemannian manifold and b a differential one-form on M with
where is the
defines a Randers metric on M and is a Randers manifold, a special case of a non-reversible Finsler manifold.[1]
Smooth quasimetric spaces
Let (M, d) be a
- 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
- The function d: M × M → [0, ∞] is smoothin some punctured neighborhood of the diagonal.
Then one can define a Finsler function F: TM →[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
and in fact any Finsler function F: TM → [0, ∞) defines an
Geodesics
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
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 [V, H] = H, where J and V are the
In analogy with the Riemannian case, there is a version
of the
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)
Notes
- .
See also
- 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
- MR 2067663
- Bao, David; MR 1747675.
- Zbl 0006.22501
- MR 1400859
- Finsler, Paul (1918), Über Kurven und Flächen in allgemeinen Räumen, Dissertation, Göttingen, JFM 46.1131.02(Reprinted by Birkhäuser (1951))
- MR 0105726.
- Shen, Zhongmin (2001). Lectures on Finsler geometry. Singapore: World Scientific. MR 1845637.