Finsler Space
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, particularly
differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...
, a Finsler manifold is a
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ...
where a (possibly asymmetric) Minkowski norm is provided on each tangent space , that enables one to define the length of any
smooth curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
as :L(\gamma) = \int_a^b F\left(\gamma(t), \dot(t)\right)\,\mathrmt. Finsler manifolds are more general than
Riemannian manifold In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
s since the tangent norms need not be induced by
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, ofte ...
s. Every Finsler manifold becomes an
intrinsic In science and engineering, an intrinsic property is a property of a specified subject that exists itself or within the subject. An extrinsic property is not essential or inherent to the subject that is being characterized. For example, mass i ...
quasimetric space when the distance between two points is defined as the infimum length of the curves that join them. named Finsler manifolds after
Paul Finsler Paul Finsler (born 11 April 1894, in Heilbronn, Germany, died 29 April 1970 in Zurich, Switzerland) was a German and Swiss mathematician. Finsler did his undergraduate studies at the Technische Hochschule Stuttgart, and his graduate studies at ...
, who studied this geometry in his dissertation .


Definition

A Finsler manifold is a
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ...
together with a Finsler metric, which is a continuous nonnegative function defined on the
tangent bundle A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Formally, in differential geometry, the tangent bundle of a differentiable manifold M is ...
so that for each point of , * for every two vectors tangent to at (
subadditivity In mathematics, subadditivity is a property of a function that states, roughly, that evaluating the function for the sum of two elements of the domain always returns something less than or equal to the sum of the function's values at each element ...
). * for all (but not necessarily for  (
positive homogeneity In mathematics, a homogeneous function is a function of several variables such that the following holds: If each of the function's arguments is multiplied by the same scalar, then the function's value is multiplied by some power of this scalar; ...
). * unless (
positive definiteness In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular: * Positive-definite bilinear form * Positive-definite f ...
). In other words, is an
asymmetric norm In mathematics, an asymmetric norm on a vector space is a generalization of the concept of a norm. Definition An asymmetric norm on a real vector space X is a function p : X \to Subadditivity">, +\infty) that has the following properties: * Su ...
on each tangent space . The Finsler metric is also required to be smooth, more precisely: * is
smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebrai ...
on the complement of the zero section of . The subadditivity axiom may then be replaced by the following strong convexity condition: * For each tangent vector , the
Hessian matrix In mathematics, the Hessian matrix, Hessian or (less commonly) Hesse matrix is a square matrix of second-order partial derivatives of a scalar-valued Function (mathematics), function, or scalar field. It describes the local curvature of a functio ...
of at is
positive definite In mathematics, positive definiteness is a property of any object to which a bilinear form In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of w ...
. Here the Hessian of at is the
symmetric Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
bilinear form In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called '' scalars''). In other words, a bilinear form is a function that is linea ...
:\mathbf_v(X, Y) := \frac\left.\frac\left (v + sX + tY)^2\right_, also known as the fundamental tensor of at . Strong convexity of implies the subadditivity with a strict inequality if . If is strongly convex, then it is a Minkowski norm on each tangent space. A Finsler metric is reversible if, in addition, * for all tangent vectors ''v''. A reversible Finsler metric defines a
norm Norm, the Norm or NORM may refer to: In academic disciplines * Normativity, phenomenon of designating things as good or bad * Norm (geology), an estimate of the idealised mineral content of a rock * Norm (philosophy), a standard in normative e ...
(in the usual sense) on each tangent space.


Examples

* Smooth submanifolds (including open subsets) of a
normed vector space The Ateliers et Chantiers de France (ACF, Workshops and Shipyards of France) was a major shipyard that was established in Dunkirk, France, in 1898. The shipyard boomed in the period before World War I (1914–18), but struggled in the inter-war ...
of finite dimension are Finsler manifolds if the norm of the vector space is smooth outside the origin. *
Riemannian manifold In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
s (but not
pseudo-Riemannian manifold In mathematical physics, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...
s) are special cases of Finsler manifolds.


Randers manifolds

Let (M, a) be a
Riemannian manifold In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
and ''b'' a differential one-form on ''M'' with :\, b\, _a := \sqrt < 1, where \left(a^\right) is the
inverse matrix In linear algebra, an invertible matrix (''non-singular'', ''non-degenarate'' or ''regular'') is a square matrix that has an inverse. In other words, if some other matrix is multiplied by the invertible matrix, the result can be multiplied by an ...
of (a_) and the
Einstein notation In mathematics, especially the usage of linear algebra in mathematical physics and differential geometry, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies ...
is used. Then :F(x, v) := \sqrt + 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 In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are a general setting for s ...
so that ''M'' is also a
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ...
and ''d'' is compatible with the
differential structure In mathematics, an ''n''- dimensional differential structure (or differentiable structure) on a set ''M'' makes ''M'' into an ''n''-dimensional differential manifold, which is a topological manifold with some additional structure that allows for ...
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'' *: \frac\, \phi(y) - \phi(x)\, \leq d(x, y) \leq C\, \phi(y) - \phi(x)\, . * The function ''d'': ''M'' × ''M'' →  , ∞is
smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebrai ...
in some punctured neighborhood of the diagonal. Then one can define a Finsler function ''F'': ''TM'' → , ∞by :F(x, v) := \lim_ \frac, 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 In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are a general setting for s ...
can be recovered from :d_L(x, y) := \inf\left\, and in fact any Finsler function ''F'': T''M'' → [0, ∞) defines an
intrinsic In science and engineering, an intrinsic property is a property of a specified subject that exists itself or within the subject. An extrinsic property is not essential or inherent to the subject that is being characterized. For example, mass i ...
quasimetric In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are a general setting for s ...
''d''''L'' on ''M'' by this formula.


Geodesics

Due to the homogeneity of ''F'' the length :L
gamma Gamma (; uppercase , lowercase ; ) is the third letter of the Greek alphabet. In the system of Greek numerals it has a value of 3. In Ancient Greek, the letter gamma represented a voiced velar stop . In Modern Greek, this letter normally repr ...
:= \int_a^b F\left(\gamma(t), \dot(t)\right)\, dt of a differentiable curve ''γ'': [''a'', ''b''] → ''M'' in ''M'' is invariant under positively oriented parametrization (geometry), reparametrizations. A constant speed curve ''γ'' is a
geodesic In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a conn ...
of a Finsler manifold if its short enough segments ''γ'', 'c'',''d''/sub> 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
gamma Gamma (; uppercase , lowercase ; ) is the third letter of the Greek alphabet. In the system of Greek numerals it has a value of 3. In Ancient Greek, the letter gamma represented a voiced velar stop . In Modern Greek, this letter normally repr ...
:= \frac\int_a^b F^2\left(\gamma(t), \dot(t)\right)\, dt in the sense that its
functional derivative In the calculus of variations, a field of mathematical analysis, the functional derivative (or variational derivative) relates a change in a functional (a functional in this sense is a function that acts on functions) to a change in a function on ...
vanishes among differentiable curves with fixed endpoints and .


Canonical spray structure on a Finsler manifold

The
Euler–Lagrange equation In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered ...
for the energy functional ''E'' 'γ''reads in the local coordinates (''x''1, ..., ''x''n, ''v''1, ..., ''v''n) of T''M'' as : g_\Big(\gamma(t), \dot\gamma(t)\Big)\ddot\gamma^i(t) + \left( \frac\Big(\gamma(t), \dot\gamma(t)\Big) - \frac\frac\Big(\gamma(t), \dot\gamma(t)\Big) \right) \dot\gamma^i(t)\dot\gamma^j(t) = 0, where ''k'' = 1, ..., ''n'' and ''g''ij is the coordinate representation of the fundamental tensor, defined as : g_(x,v) := g_v\left(\left.\frac\_x, \left.\frac\_x\right). Assuming the
strong convexity In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of the function lies above or on the graph between the two points. Equivalently, a function is convex if its ''epigraph ...
of ''F''2(''x'', ''v'') with respect to ''v'' ∈ T''x''''M'', the matrix ''g''''ij''(''x'', ''v'') is invertible and its inverse is denoted by ''g''''ij''(''x'', ''v''). Then is a geodesic of (''M'', ''F'') if and only if its tangent curve is an
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. Name Integral curves are known by various other names, depending on the nature and interpre ...
of the smooth vector field ''H'' on T''M''∖ locally defined by : \left.H\_ := \left.v^i\frac\_\!\! - \left.2G^i(x, v)\frac\_, where the local spray coefficients ''G''i are given by : G^i(x, v) := \fracg^(x, v)\left(2\frac(x, v) - \frac(x, v)\right)v^k v^\ell. The vector field ''H'' on T''M''∖ satisfies ''JH'' = ''V'' and 'V'', ''H''nbsp;= ''H'', where ''J'' and ''V'' are the canonical endomorphism and the
canonical vector field A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Formally, in differential geometry, the tangent bundle of a differentiable manifold M is a m ...
on T''M''∖. Hence, by definition, ''H'' is a
spray Spray or spraying commonly refer to: * Spray (liquid drop) ** Aerosol spray ** Blood spray ** Hair spray ** Nasal spray ** Pepper spray ** PAVA spray ** Road spray or tire spray, road debris kicked up from a vehicle tire ** Sea spray, refers to ...
on ''M''. The spray ''H'' defines a nonlinear connection on the
fibre bundle In mathematics, and particularly topology, a fiber bundle ( ''Commonwealth English'': fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
through the vertical projection :v: T(\mathrmM \setminus \) \to T(\mathrmM \setminus \);\quad v := \frac\big(I + \mathcal_H J\big). In analogy with the Riemannian case, there is a version :D_D_X(t) + R_\left(\dot\gamma(t), X(t)\right) = 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 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, who published it in 1931. Stefan Cohn-Vossen extended part of the Hopf–Rinow the ...
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 ''F''2 there exists a unique maximal geodesic ''γ'' with ''γ''(0) = x and ''γ(0) = v for any (''x'', ''v'') ∈ T''M''∖ by the uniqueness of
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. Name Integral curves are known by various other names, depending on the nature and interpre ...
s. If ''F''2 is strongly convex, geodesics ''γ'':  , ''b''nbsp;→ ''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


See also

* * *
Global analysis In mathematics, global analysis, also called analysis on manifolds, is the study of the global and topological properties of differential equations on manifolds and vector bundles. Global analysis uses techniques in infinite-dimensional manifold ...
– which uses Hilbert manifolds and other kinds of infinite-dimensional manifolds *


References

* * * * * (Reprinted by Birkhäuser (1951)) * *


External links

*
The (New) Finsler Newsletter
{{Riemannian geometry Differential geometry Finsler geometry Riemannian geometry Riemannian manifolds Smooth manifolds