In
mathematics, particularly
differential geometry, a Finsler manifold is a
differentiable manifold where a (possibly
asymmetric)
Minkowski functional 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
:
Finsler manifolds are more general than
Riemannian manifolds 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, often ...
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 ...
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 together with a Finsler metric, which is a continuous nonnegative function defined on the
tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
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, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the ''d ...
).
* 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 , +\infty) that has the following properties:
* Subadditivity, or ...
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 or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix was developed ...
of at is
positive definite 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 ...
.
Here the Hessian of at is the
symmetric
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
bilinear form
:
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
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envi ...
(in the usual sense) on each tangent space.
Examples
* Smooth submanifolds (including open subsets) of a
normed vector space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length ...
of finite dimension are Finsler manifolds if the norm of the vector space is smooth outside the origin.
*
Riemannian manifolds (but not
pseudo-Riemannian manifold
In differential geometry, 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
be a
Riemannian manifold and ''b'' a
differential one-form on ''M'' with
:
where
is the
inverse matrix
In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that
:\mathbf = \mathbf = \mathbf_n \
where denotes the -by- identity matrix and the multiplicati ...
of
and the
Einstein notation is used. Then
:
defines a Randers metric on ''M'' and
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 the most general settin ...
so that ''M'' is also a
differentiable manifold 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 dif ...
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'' →
, ∞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
:
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 the most general settin ...
can be recovered from
:
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 ...
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 the most general settin ...
''d''
''L'' on ''M'' by this formula.
Geodesics
Due to the homogeneity of ''F'' the length
:
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 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
:
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
:
where ''k'' = 1, ..., ''n'' and ''g''ij is the coordinate representation of the fundamental tensor, defined as
:
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 interpret ...
of the smooth vector field ''H'' on T''M''∖ locally defined by
:
where the local spray coefficients ''G''i are given by
:
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 (or, in 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 ...
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.
Uniqueness and minimizing properties of geodesics
By Hopf–Rinow theorem
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 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 interpret ...
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
References
*
*
*
*
* (Reprinted by Birkhäuser (1951))
*
*
External links
*
The (New) Finsler Newsletter
{{Riemannian geometry
Differential geometry
Finsler geometry
Riemannian geometry
Riemannian manifolds
Smooth manifolds