Differentiable manifold
In mathematics, a differentiable manifold is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts. One may then apply ideas from calculus while working within the individual charts, since each chart lies within a vector space to which the usual rules of calculus apply. If the charts are suitably compatible, then computations done in one chart are valid in any other differentiable chart.
In formal terms, a differentiable manifold is a topological manifold with a globally defined differential structure. Any topological manifold can be given a differential structure locally by using the homeomorphisms in its atlas and the standard differential structure on a vector space. To induce a global differential structure on the local coordinate systems induced by the homeomorphisms, their compositions on chart intersections in the atlas must be differentiable functions on the corresponding vector space. In other words, where the domains of charts overlap, the coordinates defined by each chart are required to be differentiable with respect to the coordinates defined by every chart in the atlas. The maps that relate the coordinates defined by the various charts to one another are called transition maps.
The ability to define such a local differential structure on an abstract space allows one to extend the definition of differentiability to spaces without global coordinate systems. A locally differential structure allows one to define the globally differentiable tangent space, differentiable functions, and differentiable tensor and vector fields.
Differentiable manifolds are very important in physics. Special kinds of differentiable manifolds form the basis for physical theories such as classical mechanics, general relativity, and Yang–Mills theory. It is possible to develop a calculus for differentiable manifolds. This leads to such mathematical machinery as the exterior calculus. The study of calculus on differentiable manifolds is known as differential geometry.
"Differentiability" of a manifold has been given several meanings, including: continuously differentiable, k-times differentiable, smooth, and analytic.
History
The emergence of differential geometry as a distinct discipline is generally credited to Carl Friedrich Gauss and Bernhard Riemann. Riemann first described manifolds in his famous habilitation lecture before the faculty at Göttingen. He motivated the idea of a manifold by an intuitive process of varying a given object in a new direction, and presciently described the role of coordinate systems and charts in subsequent formal developments:The works of physicists such as James Clerk Maxwell, and mathematicians Gregorio Ricci-Curbastro and Tullio Levi-Civita led to the development of tensor analysis and the notion of covariance, which identifies an intrinsic geometric property as one that is invariant with respect to coordinate transformations. These ideas found a key application in Albert Einstein's theory of general relativity and its underlying equivalence principle. A modern definition of a 2-dimensional manifold was given by Hermann Weyl in his 1913 book on Riemann surfaces. The widely accepted general definition of a manifold in terms of an atlas is due to Hassler Whitney.
Definition
Atlases
Let be a topological space. A chart on consists of an open subset of, and a homeomorphism from to an open subset of some Euclidean space. Somewhat informally, one may refer to a chart, meaning that the image of is an open subset of, and that is a homeomorphism onto its image; in the usage of some authors, this may instead mean that is itself a homeomorphism.The presence of a chart suggests the possibility of doing differential calculus on ; for instance, if given a function and a chart on, one could consider the composition, which is a real-valued function whose domain is an open subset of a Euclidean space; as such, if it happens to be differentiable, one could consider its partial derivatives.
This situation is not fully satisfactory for the following reason. Consider a second chart on, and suppose that and contain some points in common. The two corresponding functions and are linked in the sense that they can be reparametrized into one another:
the natural domain of the right-hand side being. Since and are homeomorphisms, it follows that is a homeomorphism from to. Consequently, it's just a bicontinuous function, thus even if both functions and are differentiable, their differential properties will not necessarily be strongly linked to one another, as is not guaranteed to be sufficiently differentiable for being able to compute the partial derivatives of the LHS applying the chain rule to the RHS. The same problem is found if one considers instead functions ; one is led to the reparametrization formula
at which point one can make the same observation as before.
This is resolved by the introduction of a "differentiable atlas" of charts, which specifies a collection of charts on for which the transition maps are all differentiable. This makes the situation quite clean: if is differentiable, then due to the first reparametrization formula listed above, the map is also differentiable on the region, and vice versa. Moreover, the derivatives of these two maps are linked to one another by the chain rule. Relative to the given atlas, this facilitates a notion of differentiable mappings whose domain or range is, as well as a notion of the derivative of such maps.
Formally, the word "differentiable" is somewhat ambiguous, as it is taken to mean different things by different authors; sometimes it means the existence of first derivatives, sometimes the existence of continuous first derivatives, and sometimes the existence of infinitely many derivatives. The following gives a formal definition of various meanings of "differentiable atlas". Generally, "differentiable" will be used as a catch-all term including all of these possibilities, provided.
Since every real-analytic map is smooth, and every smooth map is for any, one can see that any analytic atlas can also be viewed as a smooth atlas, and every smooth atlas can be viewed as a atlas. This chain can be extended to include holomorphic atlases, with the understanding that any holomorphic map between open subsets of can be viewed as a real-analytic map between open subsets of.
Given a differentiable atlas on a topological space, one says that a chart is differentiably compatible with the atlas, or differentiable relative to the given atlas, if the inclusion of the chart into the collection of charts comprising the given differentiable atlas results in a differentiable atlas. A differentiable atlas determines a maximal differentiable atlas, consisting of all charts which are differentiably compatible with the given atlas. A maximal atlas is always very large. For instance, given any chart in a maximal atlas, its restriction to an arbitrary open subset of its domain will also be contained in the maximal atlas. A maximal smooth atlas is also known as a smooth structure; a maximal holomorphic atlas is also known as a complex structure.
An alternative but equivalent definition, avoiding the direct use of maximal atlases, is to consider equivalence classes of differentiable atlases, in which two differentiable atlases are considered equivalent if every chart of one atlas is differentiably compatible with the other atlas. Informally, what this means is that in dealing with a smooth manifold, one can work with a single differentiable atlas, consisting of only a few charts, with the implicit understanding that many other charts and differentiable atlases are equally legitimate.
According to the invariance of domain, each connected component of a topological space which has a differentiable atlas has a well-defined dimension. This causes a small ambiguity in the case of a holomorphic atlas, since the corresponding dimension will be one-half of the value of its dimension when considered as an analytic, smooth, or atlas. For this reason, one refers separately to the "real" and "complex" dimension of a topological space with a holomorphic atlas.
Manifolds
A differentiable manifold is a Hausdorff and second countable topological space, together with a maximal differentiable atlas on. Much of the basic theory can be developed without the need for the Hausdorff and second countability conditions, although they are vital for much of the advanced theory. They are essentially equivalent to the general existence of bump functions and partitions of unity, both of which are used ubiquitously.The notion of a manifold is identical to that of a topological manifold. However, there is a notable distinction to be made. Given a topological space, it is meaningful to ask whether or not it is a topological manifold. By contrast, it is not meaningful to ask whether or not a given topological space is a smooth manifold, since the notion of a smooth manifold requires the specification of a smooth atlas, which is an additional structure. It could, however, be meaningful to say that a certain topological space cannot be given the structure of a smooth manifold. It is possible to reformulate the definitions so that this sort of imbalance is not present; one can start with a set , using the natural analogue of a smooth atlas in this setting to define the structure of a topological space on.
Patching together Euclidean pieces to form a manifold
One can reverse-engineer the above definitions to obtain one perspective on the construction of manifolds. The idea is to start with the images of the charts and the transition maps, and to construct the manifold purely from this data. As in the above discussion, we use the "smooth" context but everything works just as well in other settings.Given an indexing set let be a collection of open subsets of and for each let be an open subset of and let be a smooth map. Suppose that is the identity map, that is the identity map, and that is the identity map. Then define an equivalence relation on the disjoint union by declaring to be equivalent to With some technical work, one can show that the set of equivalence classes can naturally be given a topological structure, and that the charts used in doing so form a smooth atlas. For the patching together the analytic structures, see analytic varieties.