Talk:Manifold/old version

For other meanings of this term, see manifold (disambiguation).

In mathematics, a differentiable 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. 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. 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 an 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.

If you imagine yourself, or an ant, within a certain manifold, say the surface of Earth, you have the intrinsic view. When you step outside of the manifold, say by getting into a rocket and flying into space, and then look back at the ant, you have the extrinsic view.

Formal definition

Charts and transition maps

A K-chart at p is a homeomorphism from an open neighbourhood of p to K. Instead of saying "there is a K-chart at p", you can say "at p there is a K-chart". If at p there are two K-charts, 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 K to itself. Thus all transition maps are homeomorphisms.

Topological manifold

A topological n-manifold is a Hausdorff space in which at every point there is an Rⁿ-chart. More generally it is possible to allow a topological manifold to have a boundary which looks locally like a Euclidean half-plane. Then a topological n-manifold is a Hausdorff space in which at each point there is an Rⁿ-chart or an R₀⁺×R^n-1-chart. The set of points at which there are only R₀⁺×R^n-1-charts is called the boundary and its complement is called the interior.

Differentiable manifold

A C^k n-manifold is a topological n-manifold for which all transition maps are C^k. Thus a C⁰ n-manifold is a topological n-manifold and for k>0 we speak of differentiable manifolds.

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.

Atlas

A collection of charts which cover a manifold M is called an atlas of M. If all transition maps of an atlas are k times continuously differentiable, then the atlas is a C^k atlas. Two C^k atlases are called equivalent if their union is again a C^k atlas.

Once a C¹ atlas on a paracompact manifold is given, we can refine it to an equivalent real analytic atlas, and all such refinements give the same analytic manifold. Therefore, one often considers only analytic manifolds. If we only have a C⁰ atlas, then it cannot always be refined to an equivalent C¹ or real analytic atlas. Thus not every topological manifold is isomorphic to a differentiable manifold. Such manifolds have dimension greater than three. It is possible for two non-equivalent differentiable manifolds to be homeomorphic. A famous example of this are exotic spheres.

Why require Hausdorffness

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.

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.

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

Manifolds inherit many of the local properties of the Euclidean space. In particular, they are locally path-connected, locally compact and locally metrizable. (Readers should see the topology glossary for definitions of topological terms used in this article.) Being locally compact Hausdorff spaces they are necessarily Tychonoff spaces.

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, you can form the C^k functions to the real or complex numbers. These form an algebra under pointwise addition and multiplication, called the algebra of scalars. Its base field may be embedded into the algebra as the field of constant functions and it thus has a unit, namely 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 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 Kähler manifold which may have applications in physics.

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 use a different notion of chart known as "plots". Differential spaces and Frölicher spaces are other attempts.

References

Guillemin, Victor and Anton Pollack, Differential Topology, Prentice-Hall (1974) ISBN 0132126052. This text was inspired by Milnor, and is commonly used for undergraduate courses.
Hirsch, Morris, Differential Topology, Springer (1997) ISBN 0387901485. 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 Viewpont, Princeton University Press, (revised, 1997) ISBN 0691048339. 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 0805390219. This is the standard text used in most graduate courses.

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