In mathematics, an *n*-dimensional **differential structure** (or **differentiable structure**) on a set *M* makes *M* into an *n*-dimensional differential manifold, which is a topological manifold with some additional structure that allows for differential calculus on the manifold. If *M* is already a topological manifold, it is required that the new topology be identical to the existing one.

For a natural number *n* and some *k* which may be a non-negative integer or infinity, an *n*-dimensional *C*^{k} differential structure^{ [1] } is defined using a ** C^{k}-atlas **, which is a set of bijections called

which are ** C^{k}-compatible** (in the sense defined below):

Each such map provides a way in which certain subsets of the manifold may be viewed as being like open subsets of but the usefulness of this notion depends on to what extent these notions agree when the domains of two such maps overlap.

Consider two charts:

The intersection of the domains of these two functions is

and its map by the two chart maps to the two images:

The transition map between the two charts is the map between the two images of this intersection under the two chart maps.

Two charts are ** C^{k}-compatible** if

are open, and the transition maps

have continuous partial derivatives of order *k*. If *k* = 0, we only require that the transition maps are continuous, consequently a *C*^{0}-atlas is simply another way to define a topological manifold. If *k* = ∞, derivatives of all orders must be continuous. A family of *C*^{k}-compatible charts covering the whole manifold is a *C*^{k}-atlas defining a *C*^{k} differential manifold. Two atlases are ** C^{k}-equivalent** if the union of their sets of charts forms a

For simplification of language, without any loss of precision, one might just call a maximal *C*^{k}−atlas on a given set a *C*^{k}−manifold. This maximal atlas then uniquely determines both the topology and the underlying set, the latter being the union of the domains of all charts, and the former having the set of all these domains as a basis.

For any integer *k* > 0 and any *n*−dimensional *C*^{k}−manifold, the maximal atlas contains a *C*^{∞}−atlas on the same underlying set by a theorem due to Hassler Whitney. It has also been shown that any maximal *C*^{k}−atlas contains some number of *distinct* maximal *C*^{∞}−atlases whenever *n* > 0, although for any pair of these *distinct**C*^{∞}−atlases there exists a *C*^{∞}−diffeomorphism identifying the two. It follows that there is only one class of smooth structures (modulo pairwise smooth diffeomorphism) over any topological manifold which admits a differentiable structure, i.e. The *C*^{∞}−, structures in a *C*^{k}−manifold. A bit loosely, one might express this by saying that the smooth structure is (essentially) unique. The case for *k* = 0 is different. Namely, there exist topological manifolds which admit no *C*^{1}−structure, a result proved by Kervaire (1960),^{ [2] } and later explained in the context of Donaldson's theorem (compare Hilbert's fifth problem).

Smooth structures on an orientable manifold are usually counted modulo orientation-preserving smooth homeomorphisms. There then arises the question whether orientation-reversing diffeomorphisms exist. There is an "essentially unique" smooth structure for any topological manifold of dimension smaller than 4. For compact manifolds of dimension greater than 4, there is a finite number of "smooth types", i.e. equivalence classes of pairwise smoothly diffeomorphic smooth structures. In the case of **R**^{n} with *n* ≠ 4, the number of these types is one, whereas for *n* = 4, there are uncountably many such types. One refers to these by exotic **R**^{4}.

The following table lists the number of smooth types of the topological *m*−sphere *S*^{m} for the values of the dimension *m* from 1 up to 20. Spheres with a smooth, i.e. *C*^{∞}−differential structure not smoothly diffeomorphic to the usual one are known as exotic spheres.

Dimension | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Smooth types | 1 | 1 | 1 | ≥1 | 1 | 1 | 28 | 2 | 8 | 6 | 992 | 1 | 3 | 2 | 16256 | 2 | 16 | 16 | 523264 | 24 |

It is not currently known how many smooth types the topological 4-sphere *S*^{4} has, except that there is at least one. There may be one, a finite number, or an infinite number. The claim that there is just one is known as the *smooth* Poincaré conjecture (see * Generalized Poincaré conjecture *). Most mathematicians believe that this conjecture is false, i.e. that *S*^{4} has more than one smooth type. The problem is connected with the existence of more than one smooth type of the topological 4-disk (or 4-ball).

As mentioned above, in dimensions smaller than 4, there is only one differential structure for each topological manifold. That was proved by Tibor Radó for dimension 1 and 2, and by Edwin E. Moise in dimension 3.^{ [3] } By using obstruction theory, Robion Kirby and Laurent C. Siebenmann were able to show that the number of PL structures for compact topological manifolds of dimension greater than 4 is finite.^{ [4] } John Milnor, Michel Kervaire, and Morris Hirsch proved that the number of smooth structures on a compact PL manifold is finite and agrees with the number of differential structures on the sphere for the same dimension (see the book Asselmeyer-Maluga, Brans chapter 7) By combining these results, the number of smooth structures on a compact topological manifold of dimension not equal to 4 is finite.

Dimension 4 is more complicated. For compact manifolds, results depend on the complexity of the manifold as measured by the second Betti number *b*_{2}. For large Betti numbers *b*_{2} > 18 in a simply connected 4-manifold, one can use a surgery along a knot or link to produce a new differential structure. With the help of this procedure one can produce countably infinite many differential structures. But even for simple spaces such as one doesn't know the construction of other differential structures. For non-compact 4-manifolds there are many examples like having uncountably many differential structures.

In mathematics, a **Lie group** is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract, generic concept of multiplication and the taking of inverses (division). Combining these two ideas, one obtains a continuous group where points can be multiplied together, and their inverse can be taken. If, in addition, the multiplication and taking of inverses are defined to be smooth (differentiable), one obtains a Lie group.

In mathematics, the **tangent space** of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector that gives the displacement of the one point from the other.

In mathematics, particularly topology, one describes a manifold using an **atlas**. An atlas consists of individual *charts* that, roughly speaking, describe individual regions of the manifold. If the manifold is the surface of the Earth, then an atlas has its more common meaning. In general, the notion of atlas underlies the formal definition of a manifold and related structures such as vector bundles and other fibre bundles.

In differential geometry, a **Riemannian manifold** or **Riemannian space**(*M*, *g*) is a real, smooth manifold *M* equipped with a positive-definite inner product *g*_{p} on the tangent space *T*_{p}*M* at each point *p*. A common convention is to take *g* to be smooth, which means that for any smooth coordinate chart (*U*, *x*) on *M*, the *n*^{2} functions

In mathematics, and particularly topology, a **fiber bundle** is a space that is *locally* a product space, but *globally* may have a different topological structure. Specifically, the similarity between a space and a product space is defined using a continuous surjective map, , that in small regions of *E* behaves just like a projection from corresponding regions of to . The map , called the **projection** or **submersion** of the bundle, is regarded as part of the structure of the bundle. The space is known as the **total space** of the fiber bundle, as the **base space**, and the **fiber**.

In differential geometry, the **Ricci curvature tensor**, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure of the degree to which the geometry of a given metric tensor differs locally from that of ordinary Euclidean space or pseudo-Euclidean space.

In mathematics, a **principal bundle** is a mathematical object that formalizes some of the essential features of the Cartesian product *X* × *G* of a space *X* with a group *G*. In the same way as with the Cartesian product, a principal bundle *P* is equipped with

- An action of
*G*on*P*, analogous to (*x*,*g*)*h*= for a product space. - A projection onto
*X*. For a product space, this is just the projection onto the first factor, (*x*,*g*) ↦*x*.

In mathematics, a **submersion** is a differentiable map between differentiable manifolds whose differential is everywhere surjective. This is a basic concept in differential topology. The notion of a submersion is dual to the notion of an immersion.

In mathematics, a **foliation** is an equivalence relation on an *n*-manifold, the equivalence classes being connected, injectively immersed submanifolds, all of the same dimension *p*, modeled on the decomposition of the real coordinate space **R**^{n} into the cosets *x* + **R**^{p} of the standardly embedded subspace **R**^{p}. The equivalence classes are called the **leaves** of the foliation. If the manifold and/or the submanifolds are required to have a piecewise-linear, differentiable, or analytic structure then one defines piecewise-linear, differentiable, or analytic foliations, respectively. In the most important case of differentiable foliation of class *C ^{r}* it is usually understood that

In mathematics, **cobordism** is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary of a manifold. Two manifolds of the same dimension are *cobordant* if their disjoint union is the *boundary* of a compact manifold one dimension higher.

In mathematical analysis, the **smoothness** of a function is a property measured by the number of continuous derivatives it has over some domain. At the very minimum, a function could be considered "smooth" if it is differentiable everywhere. At the other end, it might also possess derivatives of all orders in its domain, in which case it is said to be **infinitely differentiable** and referred to as a **C-infinity function**.

In mathematics, a **smooth structure** on a manifold allows for an unambiguous notion of smooth function. In particular, a smooth structure allows one to perform mathematical analysis on the manifold.

In mathematics, a **manifold** is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or *n-manifold* for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to the Euclidean space of dimension n.

In topology, a branch of mathematics, a **topological manifold** is a topological space which locally resembles real *n*-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathematics. All manifolds are topological manifolds by definition. Other types of manifolds are formed by adding structure to a topological manifold. Every manifold has an "underlying" topological manifold, obtained by simply "forgetting" the added structure.

In mathematics, a **differentiable manifold** is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since each chart lies within a linear 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 mathematics, an **exotic** is a differentiable manifold that is homeomorphic but not diffeomorphic to the Euclidean space The first examples were found in 1982 by Michael Freedman and others, by using the contrast between Freedman's theorems about topological 4-manifolds, and Simon Donaldson's theorems about smooth 4-manifolds. There is a continuum of non-diffeomorphic differentiable structures of as was shown first by Clifford Taubes.

In mathematics, a **Banach manifold** is a manifold modeled on Banach spaces. Thus it is a topological space in which each point has a neighbourhood homeomorphic to an open set in a Banach space. Banach manifolds are one possibility of extending manifolds to infinite dimensions.

In differential geometry, in the category of differentiable manifolds, a **fibered manifold** is a surjective submersion

The **Yamabe problem** refers to a conjecture in the mathematical field of differential geometry, which was resolved in the 1980s. It is a statement about the scalar curvature of Riemannian manifolds:

Let (

M,g) be a closed smooth Riemannian manifold. Then there exists a positive and smooth function f on M such that the Riemannian metricfghas constant scalar curvature.

In geometry, if *X* is a manifold with an action of a topological group *G* by analytical diffeomorphisms, the notion of a **( G, X)-structure** on a topological space is a way to formalise it being locally isomorphic to

- ↑ Hirsch, Morris,
*Differential Topology*, Springer (1997), ISBN 0-387-90148-5. for a general mathematical account of differential structures - ↑ Kervaire, Michel (1960). "A manifold which does not admit any differentiable structure".
*Commentarii Mathematici Helvetici*.**34**: 257–270. doi:10.1007/BF02565940.CS1 maint: discouraged parameter (link) - ↑ Moise, Edwin E. (1952). "Affine structures in 3-manifolds. V. The triangulation theorem and Hauptvermutung".
*Annals of Mathematics*. Second Series.**56**(1): 96–114. doi:10.2307/1969769. JSTOR 1969769. MR 0048805.CS1 maint: discouraged parameter (link) - ↑ Kirby, Robion C.; Siebenmann, Laurence C. (1977).
*Foundational Essays on Topological Manifolds. Smoothings, and Triangulations*. Princeton, New Jersey: Princeton University Press. ISBN 0-691-08190-5.CS1 maint: discouraged parameter (link)

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