In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a sub-Riemannian manifold is a certain type of generalization of a
Riemannian manifold
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...
. Roughly speaking, to measure distances in a sub-Riemannian manifold, you are allowed to go only along curves tangent to so-called ''horizontal subspaces''.
Sub-Riemannian manifolds (and so, ''a fortiori'', Riemannian manifolds) carry a natural
intrinsic metric In the mathematical study of metric spaces, one can consider the arclength of paths in the space. If two points are at a given distance from each other, it is natural to expect that one should be able to get from the first point to the second alon ...
called the metric of Carnot–Carathéodory. The
Hausdorff dimension
In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a ...
of such
metric space
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 ...
s is always an
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
and larger than its
topological dimension
In mathematics, the Lebesgue covering dimension or topological dimension of a topological space is one of several different ways of defining the dimension of the space in a
topological invariant, topologically invariant way.
Informal discussion
F ...
(unless it is actually a Riemannian manifold).
Sub-Riemannian manifolds often occur in the study of constrained systems in
classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical ...
, such as the motion of vehicles on a surface, the motion of robot arms, and the orbital dynamics of satellites. Geometric quantities such as the
Berry phase In classical and quantum mechanics, geometric phase is a phase difference acquired over the course of a cycle, when a system is subjected to cyclic adiabatic processes, which results from the geometrical properties of the parameter space of the ...
may be understood in the language of sub-Riemannian geometry. The
Heisenberg group
In mathematics, the Heisenberg group H, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form
::\begin
1 & a & c\\
0 & 1 & b\\
0 & 0 & 1\\
\end
under the operation of matrix multiplication. Elements ' ...
, important to
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, carries a natural sub-Riemannian structure.
Definitions
By a ''distribution'' on
we mean a
subbundle of 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 ...
of
.
Given a distribution
a vector field in
is called ''horizontal''. A curve
on
is called horizontal if
for any
.
A distribution on
is called ''completely non-integrable'' if for any
we have that any tangent vector can be presented as a
linear combination of vectors of the following types
where all vector fields
are horizontal.
A sub-Riemannian manifold is a triple
, where
is a differentiable manifold,
is a completely non-integrable "horizontal" distribution and
is a smooth section of positive-definite quadratic forms on
.
Any sub-Riemannian manifold carries the natural
intrinsic metric In the mathematical study of metric spaces, one can consider the arclength of paths in the space. If two points are at a given distance from each other, it is natural to expect that one should be able to get from the first point to the second alon ...
, called the metric of Carnot–Carathéodory, defined as
:
where infimum is taken along all ''horizontal curves''
such that
,
.
Examples
A position of a car on the plane is determined by three parameters: two coordinates
and
for the location and an angle
which describes the orientation of the car. Therefore, the position of the car can be described by a point in a manifold
:
One can ask, what is the minimal distance one should drive to get from one position to another? This defines a
Carnot–Carathéodory metric
In mathematics, a sub-Riemannian manifold is a certain type of generalization of a Riemannian manifold. Roughly speaking, to measure distances in a sub-Riemannian manifold, you are allowed to go only along curves tangent to so-called ''horizontal ...
on the manifold
:
A closely related example of a sub-Riemannian metric can be constructed on a
Heisenberg group
In mathematics, the Heisenberg group H, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form
::\begin
1 & a & c\\
0 & 1 & b\\
0 & 0 & 1\\
\end
under the operation of matrix multiplication. Elements ' ...
: Take two elements
and
in the corresponding Lie algebra such that
:
spans the entire algebra. The horizontal distribution
spanned by left shifts of
and
is ''completely non-integrable''. Then choosing any smooth positive quadratic form on
gives a sub-Riemannian metric on the group.
Properties
For every sub-Riemannian manifold, there exists a
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
, called the sub-Riemannian Hamiltonian, constructed out of the metric for the manifold. Conversely, every such quadratic Hamiltonian induces a sub-Riemannian manifold. The existence of geodesics of the corresponding
Hamilton–Jacobi equation
In physics, the Hamilton–Jacobi equation, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mechan ...
s for the sub-Riemannian Hamiltonian is given by the
Chow–Rashevskii theorem
In sub-Riemannian geometry, the Chow–Rashevskii theorem (also known as Chow's theorem) asserts that any two points of a connected sub-Riemannian manifold, endowed with a bracket generating distribution, are connected by a horizontal path in the ...
.
See also
*
Carnot group In mathematics, a Carnot group is a simply connected nilpotent Lie group, together with a derivation of its Lie algebra such that the subspace with eigenvalue 1 generates the Lie algebra. The subbundle of the tangent bundle associated to this eigens ...
, a class of
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
s that form sub-Riemannian manifolds
*
Distribution Distribution may refer to:
Mathematics
*Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations
* Probability distribution, the probability of a particular value or value range of a vari ...
References
*
*
*
*
{{Riemannian geometry
Metric geometry
Riemannian geometry
Riemannian manifolds