Английская Википедия:Finsler manifold

Материал из Онлайн справочника
Перейти к навигацииПерейти к поиску

Шаблон:Redirect Шаблон:Refimprove In mathematics, particularly differential geometry, a Finsler manifold is a differentiable manifold Шаблон:Math where a (possibly asymmetric) Minkowski norm Шаблон:Math is provided on each tangent space Шаблон:Math, that enables one to define the length of any smooth curve Шаблон:Math as

<math>L(\gamma) = \int_a^b F\left(\gamma(t), \dot{\gamma}(t)\right)\,\mathrm{d}t.</math>

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.

Шаблон:Harvs named Finsler manifolds after Paul Finsler, who studied this geometry in his dissertation Шаблон:Harv.

Definition

A Finsler manifold is a differentiable manifold Шаблон:Math together with a Finsler metric, which is a continuous nonnegative function Шаблон:Math defined on the tangent bundle so that for each point Шаблон:Math of Шаблон:Math,

In other words, Шаблон:Math is an asymmetric norm on each tangent space Шаблон:Math. The Finsler metric Шаблон:Math is also required to be smooth, more precisely:

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

Here the Hessian of Шаблон:Math at Шаблон:Math is the symmetric bilinear form

<math>\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},</math>

also known as the fundamental tensor of Шаблон:Math at Шаблон:Math. Strong convexity of Шаблон:Math implies the subadditivity with a strict inequality if Шаблон:Math. If Шаблон:Math is strongly convex, then it is a Minkowski norm on each tangent space.

A Finsler metric is reversible if, in addition,

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

Examples

Randers manifolds

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

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

where <math>\left(a^{ij}\right)</math> is the inverse matrix of <math>(a_{ij})</math> and the Einstein notation is used. Then

<math>F(x, v) := \sqrt{a_{ij}(x)v^i v^j} + b_i(x)v^i</math>

defines a Randers metric on M and <math>(M, F)</math> 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
    <math> \frac{1}{C}\|\phi(y) - \phi(x)\| \leq d(x, y) \leq C\|\phi(y) - \phi(x)\|.</math>
  • 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

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

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 Шаблон:Nowrap of the original quasimetric can be recovered from

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

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

Geodesics

Due to the homogeneity of F the length

<math>L[\gamma] := \int_a^b F\left(\gamma(t), \dot{\gamma}(t)\right)\, dt</math>

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

<math>E[\gamma] := \frac{1}{2}\int_a^b F^2\left(\gamma(t), \dot{\gamma}(t)\right)\, dt</math>

in the sense that its functional derivative vanishes among differentiable curves Шаблон:Nowrap with fixed endpoints Шаблон:Nowrap and Шаблон:Nowrap.

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

<math>
 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,

</math>

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

<math>

g_{ij}(x,v) := g_v\left(\left.\frac{\partial}{\partial x^i}\right|_x, \left.\frac{\partial}{\partial x^j}\right|_x\right). </math>

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 Шаблон:Nobreak is a geodesic of (M, F) if and only if its tangent curve Шаблон:Nobreak is an integral curve of the smooth vector field H on TM∖{0} locally defined by

<math>

\left.H\right|_{(x, v)} := \left.v^i\frac{\partial}{\partial x^i}\right|_{(x,v)}\!\! - \left.2G^i(x, v)\frac{\partial}{\partial v^i}\right|_{(x,v)}, </math>

where the local spray coefficients Gi are given by

<math>

G^i(x, v) := \frac{1}{4}g^{ij}(x, v)\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. </math>

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 Шаблон:Nowrap through the vertical projection

<math>v: T(\mathrm{T}M \setminus \{0\}) \to T(\mathrm{T}M \setminus \{0\});\quad v := \frac{1}{2}\big(I + \mathcal{L}_H J\big).</math>

In analogy with the Riemannian case, there is a version

<math>D_{\dot\gamma}D_{\dot\gamma}X(t) + R_{\dot\gamma}\left(\dot\gamma(t), X(t)\right) = 0</math>

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

Шаблон:Reflist

See also

References

External links

Шаблон:Manifolds Шаблон:Riemannian geometry

Партнерские ресурсы
Криптовалюты
Магазины
Хостинг
Разное

  1. ↑ Шаблон:Cite journal
Источник — http://wikihandbk.com/ruwiki/index.php?title=Английская_Википедия:Finsler_manifold&oldid=14703801