A point of the manifold is therefore an equivalence class of points which are mapped to each other by transition maps. Charts map equivalence classes to points of a single patch. There are usually strong demands on the consistency of the transition maps. For topological manifolds they are required to be homeomorphisms ; if they are also diffeomorphisms , the resulting manifold is a differentiable manifold.
Start with two copies of the line. Use the coordinate s for the first copy, and t for the second copy. This gives a circle. The first construction and this construction are very similar, but they represent rather different points of view.
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In the first construction, the manifold is seen as 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 a Euclidean space it is always clear whether a vector at some point is tangential or normal to some surface through that point. The patchwork construction does not use any embedding, but simply views the manifold as a topological space by itself. This abstract point of view is called the intrinsic view. It can make it harder to imagine what a tangent vector might be, and there is no intrinsic notion of a normal bundle, but instead there is an intrinsic stable normal bundle.
The n -sphere S n is a generalisation of the idea of a circle 1-sphere and sphere 2-sphere to higher dimensions. An n -sphere S n can be constructed by gluing together two copies of R n. The transition map between them is defined as. This function is its own inverse and thus can be used in both directions. As the transition map is a smooth function , this atlas defines a smooth manifold. It is possible to define different points of a manifold to be same.
This can be visualized as gluing these points together in a single point, forming a quotient space. There is, however, no reason to expect such quotient spaces to be manifolds. Among the possible quotient spaces that are not necessarily manifolds, orbifolds and CW complexes are considered to be relatively well-behaved. An example of a quotient space of a manifold that is also a manifold is the real projective space identified as a quotient space of the corresponding sphere. One method of identifying points gluing them together is through a right or left action of a group , which acts on the manifold.
Two points are identified if one is moved onto the other by some group element. Manifolds which can be constructed by identifying points include tori and real projective spaces starting with a plane and a sphere, respectively. Two manifolds with boundaries can be glued together along a boundary.
If this is done the right way, the result is also a manifold. Similarly, two boundaries of a single manifold can be glued together. Formally, the gluing is defined by a bijection between the two boundaries [ dubious — discuss ]. Two points are identified when they are mapped onto each other. For a topological manifold this bijection should be a homeomorphism, otherwise the result will not be a topological manifold. Similarly for a differentiable manifold it has to be a diffeomorphism. For other manifolds other structures should be preserved.
The Cartesian product of manifolds is also a manifold. The dimension of the product manifold is the sum of the dimensions of its factors. Its topology is the product topology , and a Cartesian product of charts is a chart for the product manifold. Thus, an atlas for the product manifold can be constructed using atlases for its factors. If these atlases define a differential structure on the factors, the corresponding atlas defines a differential structure on the product manifold.
The same is true for any other structure defined on the factors. If one of the factors has a boundary, the product manifold also has a boundary. The study of manifolds combines many important areas of mathematics: it generalizes concepts such as curves and surfaces as well as ideas from linear algebra and topology. Non-Euclidean geometry considers spaces where Euclid 's parallel postulate fails.
Saccheri first studied such geometries in but sought only to disprove them. Gauss , Bolyai and Lobachevsky independently discovered them years later. Their research uncovered two types of spaces whose geometric structures differ from that of classical Euclidean space ; these gave rise to hyperbolic geometry and elliptic geometry. In the modern theory of manifolds, these notions correspond to Riemannian manifolds with constant negative and positive curvature , respectively. Carl Friedrich Gauss may have been the first to consider abstract spaces as mathematical objects in their own right.
His theorema egregium gives a method for computing the curvature of a surface without considering the ambient space in which the surface lies. Such a surface would, in modern terminology, be called a manifold; and in modern terms, the theorem proved that the curvature of the surface is an intrinsic property. Manifold theory has come to focus exclusively on these intrinsic properties or invariants , while largely ignoring the extrinsic properties of the ambient space.
Another, more topological example of an intrinsic property of a manifold is its Euler characteristic. Leonhard Euler showed that for a convex polytope in the three-dimensional Euclidean space with V vertices or corners , E edges, and F faces,. The same formula will hold if we project the vertices and edges of the polytope onto a sphere , creating a topological map with V vertices, E edges, and F faces, and in fact, will remain true for any spherical map, even if it does not arise from any convex polytope.
The Euler characteristic of other surfaces is a useful topological invariant , which can be extended to higher dimensions using Betti numbers. In the mid nineteenth century, the Gauss—Bonnet theorem linked the Euler characteristic to the Gaussian curvature. Investigations of Niels Henrik Abel and Carl Gustav Jacobi on inversion of elliptic integrals in the first half of 19th century led them to consider special types of complex manifolds , now known as Jacobians.
Bernhard Riemann further contributed to their theory, clarifying the geometric meaning of the process of analytic continuation of functions of complex variables. The possible states of a mechanical system are thought to be points of an abstract space, phase space in Lagrangian and Hamiltonian formalisms of classical mechanics. This space is, in fact, a high-dimensional manifold, whose dimension corresponds to the degrees of freedom of the system and where the points are specified by their generalized coordinates. For an unconstrained movement of free particles the manifold is equivalent to the Euclidean space, but various conservation laws constrain it to more complicated formations, e.
Liouville tori. The theory of a rotating solid body, developed in the 18th century by Leonhard Euler and Joseph-Louis Lagrange , gives another example where the manifold is nontrivial.
Manifold Theory An Introduction For Mathematical Physicists
Riemann was the first one to do extensive work generalizing the idea of a surface to higher dimensions. The name manifold comes from Riemann's original German term, Mannigfaltigkeit , which William Kingdon Clifford translated as "manifoldness". He distinguishes between stetige Mannigfaltigkeit and diskrete Mannigfaltigkeit continuous manifoldness and discontinuous manifoldness , depending on whether the value changes continuously or not. As continuous examples, Riemann refers to not only colors and the locations of objects in space, but also the possible shapes of a spatial figure.
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Riemann's intuitive notion of a Mannigfaltigkeit evolved into what is today formalized as a manifold. Riemannian manifolds and Riemann surfaces are named after Riemann. In the third section, he begins by remarking that the graph of a continuously differentiable function is a manifold in the latter sense. In this way he introduces a precursor to the notion of a chart and of a transition map. It is implicit in Analysis Situs that a manifold obtained as a 'chain' is a subset of Euclidean space. By the implicit function theorem , every submanifold of Euclidean space is locally the graph of a function.
Hermann Weyl gave an intrinsic definition for differentiable manifolds in his lecture course on Riemann surfaces in —, opening the road to the general concept of a topological space that followed shortly. During the s Hassler Whitney and others clarified the foundational aspects of the subject, and thus intuitions dating back to the latter half of the 19th century became precise, and developed through differential geometry and Lie group theory. Two-dimensional manifolds, also known as a 2D surfaces embedded in our common 3D space, were considered by Riemann under the guise of Riemann surfaces , and rigorously classified in the beginning of the 20th century by Poul Heegaard and Max Dehn.
Four-dimensional manifolds were brought to the forefront of mathematical research in the s by Michael Freedman and in a different setting, by Simon Donaldson , who was motivated by the then recent progress in theoretical physics Yang—Mills theory , where they serve as a substitute for ordinary 'flat' spacetime.
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Andrey Markov Jr. One of the most pervasive and flexible techniques underlying much work on the topology of manifolds is Morse theory. The simplest kind of manifold to define is the topological manifold, which looks locally like some "ordinary" Euclidean space R n. By definition, all manifolds are topological manifolds, so the phrase "topological manifold" is usually used to emphasize that a manifold lacks additional structure, or that only its topological properties are being considered.
Formally, a topological manifold is a topological space locally homeomorphic to a Euclidean space. This means that every point has a neighbourhood for which there exists a homeomorphism a bijective continuous function whose inverse is also continuous mapping that neighbourhood to R n. These homeomorphisms are the charts of the manifold. A topological manifold looks locally like a Euclidean space in a rather weak manner: while for each individual chart it is possible to distinguish differentiable functions or measure distances and angles, merely by virtue of being a topological manifold a space does not have any particular and consistent choice of such concepts.
In order to discuss such properties for a manifold, one needs to specify further structure and consider differentiable manifolds and Riemannian manifolds discussed below. In particular, the same underlying topological manifold can have several mutually incompatible classes of differentiable functions and an infinite number of ways to specify distances and angles.
Usually additional technical assumptions on the topological space are made to exclude pathological cases. It is customary to require that the space be Hausdorff and second countable. The dimension of the manifold at a certain point is the dimension of the Euclidean space that the charts at that point map to number n in the definition. All points in a connected manifold have the same dimension. Some authors require that all charts of a topological manifold map to Euclidean spaces of same dimension. In that case every topological manifold has a topological invariant, its dimension. Other authors allow disjoint unions of topological manifolds with differing dimensions to be called manifolds.
For most applications a special kind of topological manifold, namely a differentiable manifold , is used.