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Finsler manifold

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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

Contents

L [ γ ] = ∫ a b F ( γ ( t ) , γ ˙ ( t ) ) d t ,

where F(x, · ) is a Minkowski norm (or at least an asymmetric norm) on each tangent space TxM. 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

    g v ( X , Y ) := 1 2 ∂ 2 ∂ s ∂ t [ F ( v + s X + t Y ) 2 ] | 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 or absolutely homogenous 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 := a i j b i b j < 1 ,

    where ( a i j ) is the inverse matrix of ( a i j ) and the Einstein notation is used. Then

    F ( x , v ) := a i j ( x ) v i v 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.

    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
  • 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 → 0 + d ( γ ( 0 ) , γ ( 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 dL: M × M → [0, ∞] of the original quasimetric can be recovered from

    d L ( x , y ) := inf {   ∫ 0 1 F ( γ ( t ) , γ ˙ ( t ) ) d t   |   γ ∈ C 1 ( [ 0 , 1 ] , M )   ,   γ ( 0 ) = x   ,   γ ( 1 ) = y   } ,

    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

    L [ γ ] := ∫ a b F ( γ ( t ) , γ ˙ ( t ) ) d t

    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 [ γ ] := 1 2 ∫ a b F 2 ( γ ( t ) , γ ˙ ( t ) ) d t

    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 i k ( γ ( t ) , γ ˙ ( t ) ) γ ¨ i ( t ) + ( ∂ g i k ∂ x j ( γ ( t ) , γ ˙ ( t ) ) − 1 2 ∂ g i j ∂ x k ( γ ( t ) , γ ˙ ( t ) ) ) γ ˙ i ( t ) γ ˙ j ( t ) = 0 ,

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

    g i j ( x , v ) := g v ( ∂ ∂ x i | x , ∂ ∂ x j | x ) .

    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

    H | ( x , v ) := v i ∂ ∂ x i | ( x , v ) −   2 G i ( x , v ) ∂ ∂ v i | ( x , v ) ,

    where the local spray coefficients Gi are given by

    G i ( x , v ) := g i j ( x , v ) 4 ( 2 ∂ g j k ∂ x ℓ ( x , v ) − ∂ g k ℓ ∂ x j ( x , v ) ) v k v ℓ .

    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 ( T M ∖ 0 ) → T ( T M ∖ 0 ) ; v := 1 2 ( I + L H J ) .

    In analogy with the Riemannian case, there is a version

    D γ ˙ D γ ˙ X ( t ) + R γ ˙ ( γ ˙ ( 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.

    References

    Finsler manifold Wikipedia
    (Text) CC BY-SA