Finsler manifold
Encyclopedia
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 is given by the length functional
where F(x, · ) is a Minkowski norm (or at least an asymmetric norm
) on each tangent space
TxM. Finsler manifolds non-trivially generalize Riemannian manifold
s 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
).
named Finsler manifolds after Paul Finsler
, who studied this geometry in his dissertation .
M together with a Finsler function F defined on the tangent bundle
of M so that for all tangent vectors v,
In other words, F is an asymmetric norm
on each tangent space. Typically one replaces the subadditivity with the following strong convexity condition:
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 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 if, in addition,
A reversible Finsler metric defines a norm
(in the usual sense) on each tangent space.
and b a differential one-form
on M with
where
is the inverse matrix of 
and the Einstein notation
is used. Then
defines a Randers metric on M and (M,F) is a Randers manifold, a special case of a non-reversible Finsler manifold.
and d is compatible with the differential structure
of M in the following sense:
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 induced intrinsic metric
of the original quasimetric can be recovered from
and in fact any Finsler function F : TM →[ 0, ∞) defines an intrinsic
quasimetric dL on M by this formula.
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 with fixed endpoints γ(a)=x and γ(b)=y.
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 is a geodesic of (M,F) if and only if its tangent curve 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 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 through the vertical projection
In analogy with the Riemannian
case, there is a version
of the Jacobi equation for a general spray structure (M,H) in terms of the Ehresmann curvature
and
nonlinear covariant derivative.
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 curve
s.
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, particularly differential geometry, a Finsler manifold is a differentiable manifold
Differentiable manifold
A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...
together with the structure of an intrinsic quasimetric space in which the length of any rectifiable curve is given by the length functional
Functional
Generally, functional refers to something able to fulfill its purpose or function.*Functionalism and Functional form, movements in architectural design*Functional group, certain atomic combinations that occur in various molecules, e.g...
where F(x, · ) is a Minkowski norm (or at least an asymmetric norm
Asymmetric norm
In mathematics, an asymmetric norm on a vector space is a generalization of the concept of a norm.-Definition:Let X be a real vector space...
) on each tangent space
Tangent space
In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other....
TxM. Finsler manifolds non-trivially generalize Riemannian manifold
Riemannian manifold
In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...
s in the sense that they are not necessarily infinitesimally Euclidean
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
. This means that the (asymmetric) norm on each tangent space is not necessarily induced by an inner product (metric tensor
Metric tensor
In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...
).
named Finsler manifolds after Paul Finsler
Paul Finsler
Paul Finsler was a German and Swiss mathematician.Finsler did his undergraduate studies at the Technische Hochschule Stuttgart, and his graduate studies at the University of Göttingen, where he received his Ph.D. in 1919 under the supervision of Constantin Carathéodory...
, who studied this geometry in his dissertation .
Definition
A Finsler manifold is a differentiable manifoldDifferentiable manifold
A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...
M together with a Finsler function F defined on the tangent bundle
Tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector...
of M so that for all tangent vectors v,
- F is smoothSmooth functionIn mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...
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) (homogeneityHomogeneous functionIn mathematics, a homogeneous function is a function with multiplicative scaling behaviour: if the argument is multiplied by a factor, then the result is multiplied by some power of this factor. More precisely, if is a function between two vector spaces over a field F, and k is an integer, then...
). - 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
Asymmetric norm
In mathematics, an asymmetric norm on a vector space is a generalization of the concept of a norm.-Definition:Let X be a real vector space...
on each tangent space. Typically one replaces the subadditivity with the following strong convexity condition:
- For each tangent vector v, the hessianHessian matrixIn mathematics, the Hessian matrix is the square matrix of second-order partial derivatives of a function; that is, it describes the local curvature of a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematician Ludwig Otto Hesse and later named...
of F2 at v is 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 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 if, in addition,
- F(−v) = F(v) for all tangent vectors v.
A reversible Finsler metric defines a norm
Norm (mathematics)
In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...
(in the usual sense) on each tangent space.
Examples
- Normed vector spaceNormed vector spaceIn mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the "length" of a vector is intuitive and can easily be extended to any real vector space Rn. The following properties of "vector length" are crucial....
s of finite dimension, such as Euclidean spaceEuclidean spaceIn mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
s, whose norms are smooth outside the origin. - Riemannian manifoldRiemannian manifoldIn Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...
s (but not pseudo-Riemannian manifoldPseudo-Riemannian manifoldIn differential geometry, a pseudo-Riemannian manifold is a generalization of a Riemannian manifold. It is one of many mathematical objects named after Bernhard Riemann. The key difference between a Riemannian manifold and a pseudo-Riemannian manifold is that on a pseudo-Riemannian manifold the...
s) are special cases of Finsler manifolds.
Randers manifolds
Let (M,a) be a Riemannian manifoldRiemannian manifold
In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...
and b a differential one-form
Differential form
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...
on M with
where
is the inverse matrix of and the Einstein notationEinstein notation
In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate formulae...
is used. Then
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 manifoldDifferentiable manifold
A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...
and d is compatible with the differential structure
Differential structure
In mathematics, an n-dimensional differential structure on a set M makes M into an n-dimensional differential manifold, which is a topological manifold with some additional structure that allows us to do differential calculus on the manifold...
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 smoothSmooth functionIn mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...
in some punctured neighborhood of the diagonal.
- The function d : M × M →[0,∞[ is smooth
-
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 induced intrinsic metric
Intrinsic metric
In the mathematical study of metric spaces, one can consider the arclength of paths in the space. If two points are a given distance from each other, it is natural to expect that one should be able to get from one point to another along a path whose arclength is equal to that distance...
of the original quasimetric can be recovered from
and in fact any Finsler function F : TM →
Intrinsic metric
In the mathematical study of metric spaces, one can consider the arclength of paths in the space. If two points are a given distance from each other, it is natural to expect that one should be able to get from one point to another along a path whose arclength is equal to that distance...
quasimetric dL on M by this formula.
Geodesics
Due to the homogeneity of F the lengthof a differentiable curve γ:[a,b]→M in M is invariant under positively oriented reparametrizations
Parametrization
Parametrization is the process of deciding and defining the parameters necessary for a complete or relevant specification of a model or geometric object....
. A constant speed curve γ is a geodesic
Geodesic
In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be the shortest path between points in the space...
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
Functional derivative
In mathematics and theoretical physics, the functional derivative is a generalization of the gradient. While the latter differentiates with respect to a vector with discrete components, the former differentiates with respect to a continuous function. Both of these can be viewed as extensions of...
vanishes among differentiable curves with fixed endpoints γ(a)=x and γ(b)=y.
Canonical spray structure on a Finsler manifold
The Euler-Lagrange equationEuler-Lagrange equation
In calculus of variations, the Euler–Lagrange equation, Euler's equation, or Lagrange's equation, is a differential equation whose solutions are the functions for which a given functional is stationary...
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 is a geodesic of (M,F) if and only if its tangent curve is an integral curve
Integral curve
In mathematics, an integral curve is a parametric curve that represents a specific solution to an ordinary differential equation or system of equations...
of the smooth vector field
Vector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
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 canonical endomorphism and the canonical vector field on TM \0. Hence, by definition, H is a spray
Spray (mathematics)
In differential geometry, a spray is a vector field H on the tangent bundle TM that encodes a quasilinear second order system of ordinary differential equations on the base manifold M. Usually a spray is required to be homogeneous in the sense that its integral curves t→ΦHt∈TM obey the rule...
on M. The spray H defines a nonlinear connection
Ehresmann connection
In differential geometry, an Ehresmann connection is a version of the notion of a connection, which makes sense on any smooth fibre bundle...
on the fibre bundle through the vertical projection
Ehresmann connection
In differential geometry, an Ehresmann connection is a version of the notion of a connection, which makes sense on any smooth fibre bundle...
In analogy with the Riemannian
Riemannian manifold
In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...
case, there is a version
of the Jacobi equation for a general spray structure (M,H) in terms of the Ehresmann curvature
Ehresmann connection
In differential geometry, an Ehresmann connection is a version of the notion of a connection, which makes sense on any smooth fibre bundle...
and
nonlinear covariant derivative.
Uniqueness and minimizing properties of geodesics
By Hopf–Rinow theoremHopf–Rinow theorem
In mathematics, the Hopf–Rinow theorem is a set of statements about the geodesic completeness of Riemannian manifolds. It is named after Heinz Hopf and his student Willi Rinow .-Statement of the 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 curve
Integral curve
In mathematics, an integral curve is a parametric curve that represents a specific solution to an ordinary differential equation or system of equations...
s.
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
Riemannian manifold
In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...
case.