![MorinSurfaceAsSphere'sInsideVersusOutside](https://upload.wikimedia.org/wikipedia/commons/7/73/MorinSurfaceAsSphere%27sInsideVersusOutside.PNG)
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 ...
, the homotopy principle (or h-principle) is a very general way to solve
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...
s (PDEs), and more generally
partial differential relations (PDRs). The h-principle is good for
underdetermined PDEs or PDRs, such as the immersion problem, isometric immersion problem, fluid dynamics, and other areas.
The theory was started by
Yakov Eliashberg
Yakov Matveevich Eliashberg (also Yasha Eliashberg; russian: link=no, Яков Матвеевич Элиашберг; born 11 December 1946) is an American mathematician who was born in Leningrad, USSR.
Education and career
Eliashberg receiv ...
,
Mikhail Gromov and Anthony V. Phillips. It was based on earlier results that reduced partial differential relations to
homotopy
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...
, particularly for immersions. The first evidence of h-principle appeared in the
Whitney–Graustein theorem In the mathematical field of topology, a regular homotopy refers to a special kind of homotopy between immersions of one manifold in another. The homotopy must be a 1-parameter family of immersions.
Similar to homotopy classes, one defines two imm ...
. This was followed by the Nash–Kuiper isometric ''C''
1 embedding theorem and the Smale–Hirsch immersion theorem.
Rough idea
Assume we want to find a function ''ƒ'' on R
''m'' which satisfies a partial differential equation of degree ''k'', in co-ordinates
. One can rewrite it as
:
where
stands for all partial derivatives of ''ƒ'' up to order ''k''. Let us exchange every variable in
for new independent variables
Then our original equation can be thought as a system of
:
and some number of equations of the following type
:
A solution of
:
is called a non-holonomic solution, and a solution of the system which is also solution of our original PDE is called a holonomic solution.
In order to check whether a solution to our original equation exists, one can first check if there is a non-holonomic solution. Usually this is quite easy, and if there is no non-holonomic solution, then our original equation did not have any solutions.
A PDE ''satisfies the h-principle'' if any non-holonomic solution can be
deformed into a holonomic one in the class of non-holonomic solutions. Thus in the presence of h-principle, a differential topological problem reduces to an algebraic topological problem. More explicitly this means that apart from the topological obstruction there is no other obstruction to the existence of a holonomic solution. The topological problem of finding a ''non-holonomic solution'' is much easier to handle and can be addressed with the obstruction theory for topological bundles.
Many underdetermined partial differential equations satisfy the h-principle. However, the falsity of an h-principle is also an interesting statement, intuitively this means the objects being studied have non-trivial geometry that cannot be reduced to topology. As an example, embedded
Lagrangians in a symplectic manifold do not satisfy an h-principle, to prove this one needs to find invariants coming from
pseudo-holomorphic curves.
Simple examples
Monotone functions
Perhaps the simplest partial differential relation is for the derivative to not vanish:
Properly, this is an ''ordinary'' differential relation, as this is a function in one variable.
A holonomic solution to this relation is a function whose derivative is nowhere vanishing, i.e. a strictly monotone differentiable function, either increasing or decreasing. The space of such functions consists of two disjoint
convex set
In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex r ...
s: the increasing ones and the decreasing ones, and has the homotopy type of two points.
A non-holonomic solution to this relation would consist in the data of two functions, a differentiable function f(x), and a continuous function g(x), with g(x) nowhere vanishing. A holonomic solution gives rise to a non-holonomic solution by taking g(x) = f'(x). The space of non-holonomic solutions again consists of two disjoint convex sets, according as g(x) is positive or negative.
Thus the inclusion of holonomic into non-holonomic solutions satisfies the h-principle.
![Winding Number Around Point](https://upload.wikimedia.org/wikipedia/commons/8/8f/Winding_Number_Around_Point.svg)
This trivial example has nontrivial generalizations:
extending this to immersions of a circle into itself classifies them by order (or
winding number
In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point, i.e., the curve's number of turn ...
), by lifting the map to the
universal covering space A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties.
Definition
Let X be a topological space. A covering of X is a continuous map
: \pi : E \rightarrow X
such that there exists a discrete spa ...
and applying the above analysis to the resulting monotone map – the linear map corresponds to multiplying angle:
(
in complex numbers). Note that here there are no immersions of order 0, as those would need to turn back on themselves. Extending this to circles immersed in the plane – the immersion condition is precisely the condition that the derivative does not vanish – the
Whitney–Graustein theorem In the mathematical field of topology, a regular homotopy refers to a special kind of homotopy between immersions of one manifold in another. The homotopy must be a 1-parameter family of immersions.
Similar to homotopy classes, one defines two imm ...
classified these by
turning number
In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point, i.e., the curve's number of tu ...
by considering the homotopy class of the
Gauss map
In differential geometry, the Gauss map (named after Carl F. Gauss) maps a surface in Euclidean space R3 to the unit sphere ''S''2. Namely, given a surface ''X'' lying in R3, the Gauss map is a continuous map ''N'': ''X'' → ''S''2 such that '' ...
and showing that this satisfies an h-principle; here again order 0 is more complicated.
Smale's classification of immersions of spheres as the homotopy groups of
Stiefel manifold In mathematics, the Stiefel manifold V_k(\R^n) is the set of all orthonormal ''k''-frames in \R^n. That is, it is the set of ordered orthonormal ''k''-tuples of vectors in \R^n. It is named after Swiss mathematician Eduard Stiefel. Likewise one c ...
s, and Hirsch's generalization of this to immersions of manifolds being classified as homotopy classes of maps of
frame bundle
In mathematics, a frame bundle is a principal fiber bundle F(''E'') associated to any vector bundle ''E''. The fiber of F(''E'') over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E'x''. The general linear group acts natur ...
s are much further-reaching generalizations, and much more involved, but similar in principle – immersion requires the derivative to have rank ''k,'' which requires the partial derivatives in each direction to not vanish and to be linearly independent, and the resulting analog of the Gauss map is a map to the Stiefel manifold, or more generally between frame bundles.
A car in the plane
As another simple example, consider a car moving in the plane. The position of a car in the plane is determined by three parameters: two coordinates
and
for the location (a good choice is the location of the midpoint between the back wheels) and an angle
which describes the orientation of the car. The motion of the car satisfies the equation
:
since a non-skidding car must move in the direction of its wheels. In
robotics
Robotics is an interdisciplinary branch of computer science and engineering. Robotics involves design, construction, operation, and use of robots. The goal of robotics is to design machines that can help and assist humans. Robotics integrat ...
terms, not all paths in the task space are
holonomic.
A non-holonomic solution in this case, roughly speaking, corresponds to a motion of the car by sliding in the plane. In this case the non-holonomic solutions are not only
homotopic
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...
to holonomic ones but also can be arbitrarily well approximated by the holonomic ones (by going back and forth, like parallel parking in a limited space) – note that this approximates both the position and the angle of the car arbitrarily closely. This implies that, theoretically, it is possible to parallel park in any space longer than the length of your car. It also implies that, in a contact 3 manifold, any curve is
-close to a
Legendrian curve.
This last property is stronger than the general h-principle; it is called the
-dense h-principle.
While this example is simple, compare to the
Nash embedding theorem
The Nash embedding theorems (or imbedding theorems), named after John Forbes Nash Jr., state that every Riemannian manifold can be isometrically embedded into some Euclidean space. Isometric means preserving the length of every path. For instan ...
, specifically the
Nash–Kuiper theorem
The Nash embedding theorems (or imbedding theorems), named after John Forbes Nash Jr., state that every Riemannian manifold can be isometrically embedded into some Euclidean space. Isometric means preserving the length of every path. For insta ...
, which says that any
short
Short may refer to:
Places
* Short (crater), a lunar impact crater on the near side of the Moon
* Short, Mississippi, an unincorporated community
* Short, Oklahoma, a census-designated place
People
* Short (surname)
* List of people known as ...
smooth (
) embedding or immersion of
in
or larger can be arbitrarily well approximated by an isometric
-embedding (respectively, immersion). This is also a dense h-principle, and can be proven by an essentially similar "wrinkling" – or rather, circling – technique to the car in the plane, though it is much more involved.
Ways to prove the h-principle
*Removal of Singularities technique developed by Gromov and Eliashberg
*Sheaf technique based on the work of Smale and Hirsch.
*Convex integration based on the work of Nash and Kuiper.
Some paradoxes
Here we list a few counter-intuitive results which can be proved by applying the
h-principle:
*Cone eversion.
[D. Fuchs, S. Tabachnikov, ''Mathematical Omnibus: Thirty Lectures on Classic Mathematics''] Consider functions ''f'' on R
2 without origin ''f''(''x'') = , ''x'', . Then there is a continuous one-parameter family of functions
such that
,
and for any
,
is not zero at any point.
*Any open manifold admits a (non-complete) Riemannian metric of positive (or negative) curvature.
*
Sphere eversion
In differential topology, sphere eversion is the process of turning a sphere inside out in a three-dimensional space (the word '' eversion'' means "turning inside out"). Remarkably, it is possible to smoothly and continuously turn a sphere in ...
without creasing or tearing can be done using
immersions of
.
*The
Nash-Kuiper C1 isometric embedding theorem, in particular implies that there is a
isometric immersion of the round
into an arbitrarily small ball of
. This immersion cannot be
because a small oscillating sphere would provide a large lower bound for the principal curvatures, and therefore for the
Gauss curvature
In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point:
K = \kappa_1 \kappa_2.
The Gaussian radius of curvature is the reciprocal of .
F ...
of the immersed sphere, but on the other hand if the immersion is
this has to be equal to 1 everywhere, the Gauss curvature of the standard
, by Gauss'
Theorema Egregium
Gauss's ''Theorema Egregium'' (Latin for "Remarkable Theorem") is a major result of differential geometry, proved by Carl Friedrich Gauss in 1827, that concerns the curvature of surfaces. The theorem says that Gaussian curvature can be determi ...
.
References
Further reading
* Masahisa Adachi
Embeddings and immersions translation Kiki Hudson
*
*
*
{{DEFAULTSORT:Homotopy Principle
Partial differential equations
Mathematical principles