Talk:Manifold/old version

Historical note

This is an old version of the manifold article. It has been decided in June 2005 to rewrite this article. For details, see talk:manifold/old, talk:manifold/rewrite/freezer, and talk:manifold.

This page is about a higher mathematics topic. For other meanings of the word manifold, see manifold (disambiguation).

In mathematics, a manifold is a topological space that looks locally like the Euclidean space Rⁿ, and the Euclidean space indeed provides the simplest example of a manifold. The surface of a sphere such as the Earth provides a more complicated example. A general manifold can be obtained by bending and gluing together flat regions.

Manifolds are used in mathematics to describe geometrical objects and they provide the natural arena to study differentiability. In physics, manifolds serve as the phase space in classical mechanics and four-dimensional pseudo-Riemannian manifolds are used to model the spacetime in general relativity. They also occur as configuration spaces. The torus is the configuration space of the double pendulum.

History

The first to have conceived clearly of curves and surfaces as spaces by themselves was possibly Carl Friedrich Gauss, the founder of intrinsic differential geometry with his theorema egregium ('remarkable theorem'). Bernhard Riemann was the first to do extensive work that really required a generalization of manifolds to higher dimensions. Abelian varieties were at that time already implicitly known, as complex manifolds. Lagrangian mechanics and Hamiltonian mechanics, when considered geometrically, are also naturally manifold theories.

Intrinsic and extrinsic view

Every real manifold can be embedded in some Euclidean space. That has been proven by Hassler Whitney in the 1930s. Whitney even gave accurate bounds on dimensions — a manifold of dimension $n$ can be embedded in Euclidean space of dimension $2n$ . This is the extrinsic view. When a manifold is viewed in this way, it is easy to use intuition from Euclidean spaces to define additional structure. For example, in a Euclidean space it is always clear whether a vector at some point is tangential or normal to some surface through that point.

When we view a manifold simply as a topological space without any embedding, then it is much harder to imagine what a tangent vector might be. This is the intrinsic view. An ant on a 2-dimensional manifold, say the surface of Earth, has the intrinsic view. A space ship seeing the Earth's surface has the extrinsic view.

Technical description

In mathematics, a manifold is a topological space that looks locally like the "ordinary" Euclidean space Rⁿ and is a Hausdorff space. To make precise the notion of "looks locally like" one uses local coordinate systems or charts. A connected manifold has a definite topological dimension, which equals the number of coordinates needed in each local coordinate system. The foundational aspects of the subject were clarified during the 1930s, making precise intuitions dating back to the latter half of the 19th century, and developed through differential geometry and Lie group theory.

If the local charts on a manifold are compatible in a certain sense, one can talk about directions, tangent spaces, and differentiable functions on that manifold. These manifolds are called differentiable. In order to measure lengths and angles, even more structure is needed: one defines Riemannian manifolds to recover these geometrical ideas.

Charts and transition maps

A K-chart at p is a homeomorphism from an open neighbourhood of p to K. Usually, K is taken to be an open subset of Rⁿ. If at p there are two charts, a K₁-chart and a K₂-chart, then by restricting them to the intersection of their domains we can compose the inverse of one with the other to form a transition map from an open subset of K₁ to an open subset of K₂ -- in other words, from an open subset of Rⁿ to another open subset. All transition maps are continuous (as compositions and restrictions of continuous maps), and since the inverse of a transition map is also a transition map (by inverting the roles of K₁ and K₂), all transition maps are homeomorphisms. The definition of a manifold implies that for every p in the manifold, there exists a chart.

Topological manifolds

Topological manifold without boundary

The prototypical example of a topological manifold without boundary is Euclidean space. A general manifold without boundary looks locally, as a topological space, like Euclidean space. This is formalized by requiring that a manifold without boundary is a non-empty topological space in which every point has an open neighbourhood homeomorphic to (an open subset of) Rⁿ (Euclidean n-space). Another way of saying this, using charts, is that a manifold without boundary is a non-empty topological space in which at every point there is an Rⁿ-chart.

Topological manifold with boundary

More generally it is possible to allow a topological manifold to have a boundary. The prototypical example of a topological manifold with boundary is the Euclidean closed half-space. Most points in Euclidean closed half-space, those not on the boundary, have a neighbourhood homeomorphic to Euclidean space in addition to having a neighbourhood homeomorphic to Euclidean closed half-space, but the points on the boundary only have neighbourhoods homeomorphic to Euclidean closed half-space and not to Euclidean space. Thus we need to allow for two kinds of points in our topological manifold with boundary: points in the interior and points in the boundary. Points in the interior will, as before, have neighbourhoods homeomorphic to Euclidean space, but may also have neighbourhoods homeomorphic to Euclidean closed half-space. Points in the boundary will have neighbourhoods homeomorphic to Euclidean closed half-space. Thus a topological manifold with boundary is a non-empty topological space in which at each point there is an Rⁿ-chart or an [0,∞)×Rⁿ⁻¹-chart. The set of points at which there are only [0,∞)×Rⁿ⁻¹-charts is called the boundary and its complement is called the interior. The interior is always non-empty and is a topological n-manifold without boundary. If the boundary is non-empty then it is a topological (n-1)-manifold without boundary. If the boundary is empty, then we regain the definition of a topological manifold without boundary.

Examples

The open interval (0,1) is a one-dimensional manifold without boundary. The closed interval [0,1] is a one-dimensional manifold with boundary. Every connected one-dimensional manifold is homeomorphic to one or the other of these. The closed unit disk is a two-dimensional manifold with boundary. The plane R² and the sphere S² are two-dimensional manifolds without boundary. They are not homeomorphic, since the former is non-compact and the latter compact. The torus T² and the projective plane P² are other examples of compact two-dimensional manifolds without boundary. The projective plane is an example of a non-orientable manifold. Every compact, connected two-manifold is homeomorphic to a sphere, to a torus, to a connected sum of torii, or to a connected sum of torii and one projective plane. This is the solution to the classification problem for compact, connected two-manifolds. In higher dimensions, the classification problem has not yet been solved, and is an active area of mathematical research. Higher dimensional manifolds are harder to visualize, but are important in mathematics and physics. Space-time may be a four-dimensional manifold, or may have singular points (singularities) at the big bang and at black holes where no manifold structure exists. Infinite dimensional manifolds also exist, at least mathematically.

Properties

A manifold with empty boundary is said to be closed if it is compact, and open if it is not compact. See closed manifold.

Manifolds inherit many of the local properties of Euclidean space. In particular, they are locally path-connected, locally compact and locally metrizable. Being locally compact Hausdorff spaces they are necessarily Tychonoff spaces. Requiring a manifold to be Hausdorff may seem strange; it is tempting to think that being locally homeomorphic to a Euclidean space implies being a Hausdorff space. A counterexample is created by deleting zero from the real line and replacing it with two points, an open neighborhood of either of which includes all nonzero numbers in some open interval centered at zero. This construction, called the real line with two origins is not Hausdorff, because the two origins cannot be separated.

A topological space is said to be homogeneous if its homeomorphism group acts transitively on it. Every connected manifold without boundary is homogeneous, but manifolds with nonempty boundary are not homogeneous.

It can be shown that a manifold is metrizable if and only if it is paracompact. Non-paracompact manifolds (such as the long line) are generally regarded as pathological, so it's common to add paracompactness to the definition of an n-manifold. Sometimes n-manifolds are defined to be second-countable, which is precisely the condition required to ensure that the manifold embeds in some finite-dimensional Euclidean space. Note that every compact manifold is second-countable, and every second-countable manifold is paracompact.

Differentiable manifolds

It is easy to define the notion of a topological manifold, but it is very hard to work with this object. The smooth manifold defined below works better for most applications, in particular it makes possible to apply "calculus" on the manifold.

We start with a topological manifold M without boundary. An open set of M together with a homeomorphism between the open set and an open set of Rⁿ is called a coordinate chart. A collection of charts which cover M is called an atlas of M. The homeomorphisms of two overlapping charts provide a transition map from a subset of Rⁿ to some other subset of Rⁿ. If all these maps are k times continuously differentiable, then the atlas is an C^k atlas.

Example: The unit sphere in R³ can be covered by two charts: the complements of the north and south poles with coordinate maps - stereographic projections relative to the two poles.

Atlas

Two C^k atlases are called equivalent if their union is a C^k atlas. This is an equivalence relation, and a C^k manifold is defined to be a manifold together with an equivalence class of C^k atlases. If all the connecting maps are infinitely often differentiable, then one speaks of a smooth or C^∞ manifold; if they are all analytic, then the manifold is an analytic or C^ω manifold.

Intuitively, a smooth atlas provides local coordinate systems such that the change-of-coordinate functions are smooth. These coordinate systems allow one to define differentiability and integrability of functions on M.

Once a C¹ atlas on a paracompact manifold is given, we can refine it to a real analytic atlas (meaning that the new atlas, considered as a C¹ atlas, is equivalent to the given one), and all such refinements give the same analytic manifold. Therefore, one often considers only these latter manifolds.

Not every topological manifold admits such a smooth atlas. The lowest dimension is 4 where there are non-smoothable topological manifolds. Also, it is possible for two non-equivalent differentiable manifolds to be homeomorphic. The famous example was given by John Milnor of exotic 7-spheres, i.e. non-diffeomorphic topological 7-spheres.

The Hausdorff assumption

Requiring a manifold to be Hausdorff may seem strange; it is tempting to think that being locally homeomorphic to a Euclidean space implies being a Hausdorff space. A counterexample is created by deleting zero from the real line and replacing it with two points, an open neighborhood of either of which includes all nonzero numbers in some open interval centered at zero. This construction, called the real line with two origins is not Hausdorff, because the two origins cannot be separated.

Homogenous, second-countable and paracompact

A manifold is said to be homogeneous for its homeomorphism group, or diffeomorphism group, if that group acts transitively on it; this is true for connected manifolds. Thus every connected manifold without boundary is homogeneous.

It can be shown that a manifold is metrizable if and only if it is paracompact. Non-paracompact manifolds (such as the long line) are generally regarded as pathological, so it's common to add paracompactness to the definition of an n-manifold. Sometimes n-manifolds are defined to be second-countable, which is precisely the condition required to ensure that the manifold embeds in some finite-dimensional Euclidean space.

Note that every compact manifold is second-countable, and every second-countable manifold is paracompact.

Tangent space

Associated with every point on a differentiable manifold is a tangent space and its dual, the cotangent space. The former consists of the possible directional derivatives, and the latter of the differentials, which can be thought of as infinitesimal elements of the manifold. These spaces always have the same dimension n as the manifold does. The collection of all tangent spaces can in turn be made into a manifold, the tangent bundle, whose dimension is 2n.

Algebra of scalars

For a C^k manifold M, the set of real- or complex-valued C^k functions on the manifold forms an algebra under pointwise addition and multiplication, called the algebra of scalars. The unit of this algebra is the constant function 1.

Classification of manifolds

It is known that every second-countable connected 1-manifold without boundary is homeomorphic either to R or the circle. (The unconnected ones are just disjoint unions of these.)

For a classification of 2-manifolds, see Surface.

The 3-dimensional case may be solved. Thurston's Geometrization Conjecture, if true, together with current knowledge, would imply a classification of 3-manifolds. Grigori Perelman may have proven this conjecture; his work is currently being evaluated, as of June 14, 2003.

The classification of n-manifolds for n greater than three is known to be impossible; it is equivalent to the so-called word problem in group theory, which has been shown to be undecidable. In other words, there is no algorithm for deciding whether a given manifold is simply connected. However, there is a classification of simply connected manifolds of dimension ≥ 5.

Additional structures and generalizations

In order to do geometry on manifolds it is usually necessary to adorn these spaces with additional structures, such as the differential structure discussed above. There are numerous other possibilities, depending on the kind of geometry one is interested in:

A Riemannian manifold is a differentiable manifold on which the tangent spaces are equipped with inner products in a differentiable fashion. The inner product structure is given in the form of a symmetric 2-tensor called the Riemannian metric. On a Riemannian manifold one has notions of length, volume, and angle.
A pseudo-Riemannian manifold is a variant of Riemannian manifold where the metric tensor is allowed to have an indefinite signature (as opposed to a positive-definite one). Pseudo-Riemannian manifolds of signature (3, 1) are important in general relativity.
A symplectic manifold is a manifold equipped with a closed, nondegenerate, alternating 2-form. Such manifolds arise in the study of Hamiltonian mechanics.
A complex manifold is a manifold modeled on Cⁿ with holomorphic transition functions on chart overlaps. These manifolds are the basic objects of study in complex geometry.
A Kähler manifold is a manifold which simultaneously carries a Riemannian structure, a symplectic structure, and a complex structure which are all compatible in some suitable sense.
A Calabi-Yau manifold is a compact Ricci-flat Kähler manifold. In string theory the extra dimensions are curled up into a Calabi-Yau manifold.
A Finsler manifold is a generalization of a Riemannian manifold.
A Lie group is C^∞ manifold which also carries a smooth group structure. These are the proper objects for describing symmetries of analytical structures.

Manifolds "locally look like" Euclidean space Rⁿ and are therefore inherently finite-dimensional objects. To allow for infinite dimensions, one may consider Banach manifolds which locally look like Banach spaces, or Fréchet manifolds, which locally look like Fréchet spaces.

Another generalization of manifold allows one to omit the requirement that a manifold be Hausdorff. It still must be second-countable and locally Euclidean, however. Such spaces are called non-Hausdorff manifolds and are used in the study of codimension-1 foliations.

An orbifold is yet an another generalization of manifold, one that allows certain kinds of "singularities" in the topology. Roughly speaking, it is a space which locally looks like the quotient of Euclidean space by a finite group. The singularities correspond to fixed points of the group action.

The category of smooth manifolds with smooth maps lacks certain desirable properties, and people have tried to generalize smooth manifolds in order to rectify this. The diffeological spaces, (differential spaces) use a different notion of chart known as "plot". Frölicher spaces are another attempt.

References

Guillemin, Victor and Anton Pollack, Differential Topology, Prentice-Hall (1974) ISBN 0-13-212605-2. This text was inspired by Milnor, and is commonly used for undergraduate courses.
Hirsch, Morris, Differential Topology, Springer (1997) ISBN 0-387-90148-5. Hirsch gives the most complete account with historical insights and excellent, but difficult problems. This is the best reference for those wishing to have a deep understanding of the subject.
Kirby, Robion C.; Siebenmann, Laurence C. Foundational Essays on Topological Manifolds. Smoothings, and Triangulations. Princeton, New Jersey: Princeton University Press (1977). ISBN 0-691-08190-5. A detailed study of the category of topological manifolds.
Lee, John M. Introduction to Topological Manifolds, Springer-Verlag, New York (2000). ISBN 0-387-98759-2. Introduction to Smooth Manifolds, Springer-Verlag, New York (2003). ISBN 0-387-95495-3. Graduate-level textbooks on topological and smooth manifolds.
Milnor, John, Topology from the Differentiable Viewpoint, Princeton University Press, (revised, 1997) ISBN 0-691-04833-9. This short text may be the best math book ever written.
Spivak, Michael, Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. HarperCollins Publishers (June 1, 1965) ISBN 0-8053-9021-9. This is a standard text used in many undergraduate courses.

External links

Maxim Kazarian's Calculus on Manifolds - Independent Moscow University