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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 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 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, 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). * 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 other words, is an asymmetric norm on each tangent space . The Finsler metric is also required to be smooth, more precisely: * is smooth 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. Here the Hessian of at is the symmetric
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 (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 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 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 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 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 can be recovered from :d_L(x, y) := \inf\left\, and in fact any Finsler function ''F'': T''M'' → [0, ∞) defines an intrinsic quasimetric ''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 vanishes among differentiable curves with fixed endpoints and .


Canonical spray structure on a Finsler manifold

The Euler–Lagrange equation 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 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 on T''M''∖. Hence, by definition, ''H'' is a spray 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 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 – 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