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- near every point of the space, we have a coordinate system; or
- near every point, the environment is like that in Euclidean space of a given dimension.
Therefore, the Euclidean space itself gives the first example of a manifold. The surface of a sphere such as the Earth provides a more complicated example. Note that the whole surface cannot be drawn on one map, but it can be covered by just a few maps, and hence the surface of the Earth is a (two-dimensional) manifold.
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, were also naturally manifold theories, with a concept of generalized coordinates.
Intrinsic versus extrinsic
The given characterizations are intrinsic to M: if we imagine a small insect on (or maybe better "in") M, with eyes that only see nearby points, we are describing it from the insect's point of view. It is also possible and very useful to describe a manifold from the point of view of an outside observer. For example if a fly is crawling on an orange, we can watch this from outside in three-dimensional space, while the fly is staying on the two-dimension surface of orange peel. This point of view is called extrinsic. It is historically prior to the intrinsic point of view. During the nineteenth century, first geometry learned to consider that N dimensions were mathematically natural, with N > 3, and then that the intrinsic point of view was also geometrical. This was seen in a number of ways, for example when 'space' meant phase space in physics, or 'geometry' meant curvature in Riemannian geometry.
Therefore there are dual points of view to acquire on manifolds. They have a certain kind of intrinsic geometry, starting with their topology. They also have a geometry inside other spaces, an extrinsic geometry that depends on how they are 'mapped' into another space (think for example that every helix is the same line wrapped in different ways round cylinders). Manifolds include familiar curves such as the circle, or surfaces in three-dimensional space that are locally smooth. They include many other possibilities that are harder to visualise, such as the Lie groups basic to mathematics and theoretical physics.
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. What follows below is a clean, contemporary mathematical treatment of manifolds; 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.
Differentiable manifolds are used in mathematics to describe geometrical objects; they are also the most natural and general setting to study differentiability. In physics, differentiable manifolds serve as the phase space in classical mechanics and four dimensional pseudo-Riemannian manifolds are used to model spacetime in general relativity.
A topological n-manifold with boundary is a Hausdorff space in which every point has an open neighbourhood homeomorphic to either an open subset of E n (Euclidean n-space) or an open subset of the closed half of E n. The set of points which have an open neighbourhood homeomorphic to E n is called the interior of the manifold; it is always non-empty. The complement of the interior, is called the boundary; it is an (n-1)-manifold.
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 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. 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 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.
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 En 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 En to some other subset of En. If all these maps are k times continuously differentiable, then the atlas is an Ck atlas.
Two Ck atlases are called equivalent if their union is a Ck atlas. This is an equivalence relation, and a Ck manifold is defined to be a manifold together with an equivalence class of Ck 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 Cω 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.
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.
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.
Classification of manifolds
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 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.
- 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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