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As an example, Chow's theorem states that any projective complex manifold is in fact a projective variety — it has an algebraic structure. The transition maps of the atlas are the functions. Every topological manifold has an atlas. A C k -atlas is an atlas whose transition maps are C k. A topological manifold has a C 0 -atlas and in general a C k -manifold has a C k -atlas. If the atlas is at least C 1 , it is also called a differential structure or differentiable structure.
A holomorphic atlas is an atlas whose underlying Euclidean space is defined on the complex field and whose transition maps are biholomorphic.
Different atlases can give rise to, in essence, the same manifold. The circle can be mapped by two coordinate charts, but if the domains of these charts are changed slightly a different atlas for the same manifold is obtained. These different atlases can be combined into a bigger atlas. It can happen that the transition maps of such a combined atlas are not as smooth as those of the constituent atlases. If C k atlases can be combined to form a C k atlas, then they are called compatible. Compatibility of atlases is an equivalence relation ; by combining all the atlases in an equivalence class , a maximal atlas can be constructed.
Each C k atlas belongs to a unique maximal C k atlas. The notion of a pseudogroup  provides a flexible generalization of atlases in order to allow a variety of different structures to be defined on manifolds in a uniform way. These last three conditions are analogous to the definition of a group. For example, the collection of all local C k diffeomorphisms on R n form a pseudogroup.
All biholomorphisms between open sets in C n form a pseudogroup. Thus, a wide variety of function classes determine pseudogroups. A differentiable manifold is then an atlas compatible with the pseudogroup of C k functions on R n. A complex manifold is an atlas compatible with the biholomorphic functions on open sets in C n. And so forth. Thus, pseudogroups provide a single framework in which to describe many structures on manifolds of importance to differential geometry and topology.
Sometimes, it can be useful to use an alternative approach to endow a manifold with a C k -structure.
Instead of considering coordinate charts, it is possible to start with functions defined on the manifold itself. Thus, the sheaf-theoretic viewpoint is that the functions on a differentiable manifold can be expressed in local coordinates as differentiable functions on R n , and a fortiori this is sufficient to characterize the differential structure on the manifold. A similar, but more technical, approach to defining differentiable manifolds can be formulated using the notion of a ringed space.
This approach is strongly influenced by the theory of schemes in algebraic geometry , but uses local rings of the germs of differentiable functions. It is especially popular in the context of complex manifolds. We begin by describing the basic structure sheaf on R n. As U varies, this determines a sheaf of rings on R n. In particular, this is a local ring whose unique maximal ideal consists of those functions that vanish at p.
The pair R n , O is an example of a locally ringed space : it is a topological space equipped with a sheaf whose stalks are each local rings. In this way, differentiable manifolds can be thought of as schemes modelled on R n. There are a number of important motivations for studying differentiable manifolds within this abstract framework. First, there is no a priori reason that the model space needs to be R n. For example, in particular in algebraic geometry , one could take this to be the space of complex numbers C n equipped with the sheaf of holomorphic functions thus arriving at the spaces of complex analytic geometry , or the sheaf of polynomials thus arriving at the spaces of interest in complex algebraic geometry.
In broader terms, this concept can be adapted for any suitable notion of a scheme see topos theory. Second, coordinates are no longer explicitly necessary to the construction. The analog of a coordinate system is the pair f , f , but these merely quantify the idea of local isomorphism rather than being central to the discussion as in the case of charts and atlases. Third, the sheaf O M is not manifestly a sheaf of functions at all. Rather, it emerges as a sheaf of functions as a consequence of the construction via the quotients of local rings by their maximal ideals. Hence, it is a more primitive definition of the structure see synthetic differential geometry.
A final advantage of this approach is that it allows for natural direct descriptions of many of the fundamental objects of study to differential geometry and topology.
Algebraic, Linear and Differentiable Manifolds, Arithmetic and Moduli
In general, there will be many available charts; however, the definition of differentiability does not depend on the choice of chart at p. It follows from the chain rule applied to the transition functions between one chart and another that if f is differentiable in any particular chart at p , then it is differentiable in all charts at p.
Analogous considerations apply to defining C k functions, smooth functions, and analytic functions. There are various ways to define the derivative of a function on a differentiable manifold, the most fundamental of which is the directional derivative.
The definition of the directional derivative is complicated by the fact that a manifold will lack a suitable affine structure with which to define vectors. Therefore, the directional derivative looks at curves in the manifold instead of vectors. Given a real valued function f on an m dimensional differentiable manifold M , the directional derivative of f at a point p in M is defined as follows. This means that the directional derivative depends only on the tangent vector of the curve at p.
Thus, the more abstract definition of directional differentiation adapted to the case of differentiable manifolds ultimately captures the intuitive features of directional differentiation in an affine space. Therefore, the equivalence classes are curves through p with a prescribed velocity vector at p. The collection of all tangent vectors at p forms a vector space : the tangent space to M at p , denoted T p M. If X is a tangent vector at p and f a differentiable function defined near p , then differentiating f along any curve in the equivalence class defining X gives a well-defined directional derivative along X :.
This linear functional is often denoted by df p and is called the differential of f at p :. One of the topological features of the sheaf of differentiable functions on a differentiable manifold is that it admits partitions of unity. This distinguishes the differential structure on a manifold from stronger structures such as analytic and holomorphic structures that in general fail to have partitions of unity.
Every open covering of a C k manifold M has a C k partition of unity. This allows for certain constructions from the topology of C k functions on R n to be carried over to the category of differentiable manifolds. In particular, it is possible to discuss integration by choosing a partition of unity subordinate to a particular coordinate atlas, and carrying out the integration in each chart of R n. Partitions of unity therefore allow for certain other kinds of function spaces to be considered: for instance L p spaces , Sobolev spaces , and other kinds of spaces that require integration.
Suppose M and N are two differentiable manifolds with dimensions m and n , respectively, and f is a function from M to N. Since differentiable manifolds are topological spaces we know what it means for f to be continuous. We know what that means when f is a function between Euclidean spaces, so if we compose f with a chart of M and a chart of N such that we get a map that goes from Euclidean space to M to N to Euclidean space we know what it means for that map to be C k R m , R n.
Once again, the chain rule guarantees that the idea of differentiability does not depend on which charts of the atlases on M and N are selected. However, defining the derivative itself is more subtle. If M or N is itself already a Euclidean space, then we don't need a chart to map it to one. For a C k manifold M , the set of real-valued C k functions on the manifold forms an algebra under pointwise addition and multiplication, called the algebra of scalar fields or simply the algebra of scalars. This algebra has the constant function 1 as the multiplicative identity, and is a differentiable analog of the ring of regular functions in algebraic geometry.
On the converse, every maximal ideal in this algebra is an ideal of functions vanishing at a single point, which demonstrates that MSpec the Max Spec of C k M recovers M as a point set, though in fact it recovers M as a topological space. One can define various geometric structures algebraically in terms of the algebra of scalars, and these definitions often generalize to algebraic geometry interpreting rings geometrically and operator theory interpreting Banach spaces geometrically.
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For example, the tangent bundle to M can be defined as the derivations of the algebra of smooth functions on M. This is the basis of the field of noncommutative geometry. The tangent space of a point consists of the possible directional derivatives at that point, and has the same dimension n as does the manifold. The collection of tangent spaces at all points can in turn be made into a manifold, the tangent bundle , whose dimension is 2 n. The tangent bundle is where tangent vectors lie, and is itself a differentiable manifold. The Lagrangian is a function on the tangent bundle.
One can also define the tangent bundle as the bundle of 1- jets from R the real line to M. The transition maps on this atlas are defined from the transition maps on the original manifold, and retain the original differentiability class. The dual space of a vector space is the set of real valued linear functions on the vector space.
The cotangent space at a point is the dual of the tangent space at that point, and the cotangent bundle is the collection of all cotangent spaces. Like the tangent bundle, the cotangent bundle is again a differentiable manifold. The Hamiltonian is a scalar on the cotangent bundle. The total space of a cotangent bundle has the structure of a symplectic manifold.
Cotangent vectors are sometimes called covectors. One can also define the cotangent bundle as the bundle of 1- jets of functions from M to R. Elements of the cotangent space can be thought of as infinitesimal displacements: if f is a differentiable function we can define at each point p a cotangent vector df p , which sends a tangent vector X p to the derivative of f associated with X p.
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However, not every covector field can be expressed this way. Those that can are referred to as exact differentials. For a given set of local coordinates x k , the differentials dx k p form a basis of the cotangent space at p. The tensor bundle is the direct sum of all tensor products of the tangent bundle and the cotangent bundle. Each element of the bundle is a tensor field , which can act as a multilinear operator on vector fields, or on other tensor fields. The tensor bundle is not a differentiable manifold in the traditional sense, since it is infinite dimensional.
It is however an algebra over the ring of scalar functions. Each tensor is characterized by its ranks, which indicate how many tangent and cotangent factors it has. Sometimes these ranks are referred to as covariant and contravariant ranks, signifying tangent and cotangent ranks, respectively. A frame or, in more precise terms, a tangent frame , is an ordered basis of particular tangent space. Likewise, a tangent frame is a linear isomorphism of R n to this tangent space.