Differentiable manifold

In mathematics, a differentiable manifold (also differential 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 (atlas). 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 (namely, the transition from one chart to another is differentiable), 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 (which itself has many meanings), and analytic.

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:

Having constructed the notion of a manifoldness of n dimensions, and found that its true character consists in the property that the determination of position in it may be reduced to n determinations of magnitude, ..., B. Riemann

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.

Let M be a topological space. A chart (U, φ) on M consists of an open subset U of M, and a homeomorphism φ from U to an open subset of some Euclidean space R. Somewhat informally, one may refer to a chart φ : U → R, meaning that the image of φ is an open subset of R, and that φ is a homeomorphism onto its image; in the usage of some authors, this may instead mean that φ : U → R is itself a homeomorphism.

The presence of a chart suggests the possibility of doing differential calculus on M; for instance, if given a function u : M → R and a chart (U, φ) on M, one could consider the composition u ∘ φ, 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 (V, ψ) on M, and suppose that U and V contain some points in common. The two corresponding functions u ∘ φ and u ∘ ψ are linked in the sense that they can be reparametrized into one another: \[u\circ\varphi^{-1}=\big(u\circ\psi^{-1}\big)\circ\big(\psi\circ\varphi^{-1}\big),\] the natural domain of the right-hand side being φ(U ∩ V). Since φ and ψ are homeomorphisms, it follows that ψ ∘ φ is a homeomorphism from φ(U ∩ V) to ψ(U ∩ V). Consequently, it's just a bicontinuous function, thus even if both functions u ∘ φ and u ∘ ψ are differentiable, their differential properties will not necessarily be strongly linked to one another, as ψ ∘ φ 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 c : R → M; one is led to the reparametrization formula \[\varphi\circ c=\big(\varphi\circ\psi^{-1}\big)\circ\big(\psi\circ c\big),\] 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 M for which the transition maps ψ ∘ φ are all differentiable. This makes the situation quite clean: if u ∘ φ is differentiable, then due to the first reparametrization formula listed above, the map u ∘ ψ is also differentiable on the region ψ(U ∩ V), 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 M, 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 (nonambiguous) meanings of "differentiable atlas". Generally, "differentiable" will be used as a catch-all term including all of these possibilities, provided k ≥ 1.

Condensed: the full section is in Wikipedia.

A differentiable manifold is a Hausdorff and second countable topological space M, together with a maximal differentiable atlas on M. 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 C 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 (for instance) 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 M (rather than a topological space M), using the natural analogue of a smooth atlas in this setting to define the structure of a topological space on M.

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 \(A,\) let \(V_\alpha\) be a collection of open subsets of \(\mathbb{R}^n\) and for each \(\alpha,\beta \in A\) let \(V_{\alpha\beta}\) be an open (possibly empty) subset of \(V_\beta\) and let \(\phi_{\alpha\beta}:V_{\alpha\beta} \to V_{\beta\alpha}\) be a smooth map. Suppose that \(\phi_{\alpha\alpha}\) is the identity map, that \(\phi_{\alpha\beta} \circ \phi_{\beta\alpha}\) is the identity map, and that \(\phi_{\alpha\beta} \circ \phi_{\beta\gamma} \circ \phi_{\gamma\alpha}\) is the identity map. Then define an equivalence relation on the disjoint union \(\bigsqcup_{\alpha \in A} V_\alpha\) by declaring \(p \in V_{\alpha\beta}\) to be equivalent to \(\phi_{\alpha\beta}(p) \in V_{\beta\alpha}.\) 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(subset), see analytic varieties.

A real valued function f on an n-dimensional differentiable manifold M is called differentiable at a point p ∈ M if it is differentiable in any coordinate chart defined around p. In more precise terms, if \((U,\phi)\) is a differentiable chart where \(U\) is an open set in \(M\) containing p and \(\phi : U\to {\mathbf R}^n\) is the map defining the chart, then f is differentiable at p if and only if \[f\circ \phi^{-1} \colon \phi(U)\subset {\mathbf R}^n \to {\mathbf R}\] is differentiable at \(\phi(p)\), that is \(f\circ \phi^{-1}\) is a differentiable function from the open set \(\phi(U)\), considered as a subset of \({\mathbf R}^n\), to \(\mathbf R\). 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 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.

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.

Suppose that M is a manifold of class C, where 0 ≤ k ≤ ∞. Let {U_α} be an open covering of M. Then a partition of unity subordinate to the cover {U_α} is a collection of real-valued C functions φ_i on M satisfying the following conditions:

• The supports of the φ_i are compact and locally finite;
• The support of φ_i is completely contained in U_α for some α;
• The φ_i sum to one at each point of M: \[\sum_i \phi_i(x) = 1.\]

(Note that this last condition is actually a finite sum at each point because of the local finiteness of the supports of the φ_i.)

Every open covering of a C manifold M has a C partition of unity. This allows for certain constructions from the topology of C functions on R 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. Partitions of unity therefore allow for certain other kinds of function spaces to be considered: for instance L 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. But what does "f is C(M, N)" mean for k ≥ 1? 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(R, R). We define "f is C(M, N)" to mean that all such compositions of f with charts are C(R, R). 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.

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. For a set of (non-singular) coordinates x_k local to the point, the coordinate derivatives \(\partial_k=\frac{\partial}{\partial x_k}\) define a holonomic basis of the tangent space. The collection of tangent spaces at all points can in turn be made into a manifold, the tangent bundle, whose dimension is 2n. 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.

One may construct an atlas for the tangent bundle consisting of charts based on U_α × R, where U_α denotes one of the charts in the atlas for M. Each of these new charts is the tangent bundle for the charts U_α. 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 elements are referred to as cotangent vectors; the cotangent bundle is the collection of all cotangent vectors, along with the natural differentiable manifold structure.

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. 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, the differentials dx
_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 to this tangent space. A moving tangent frame is an ordered list of vector fields that give a basis at every point of their domain. One may also regard a moving frame as a section of the frame bundle F(M), a GL(n, R) principal bundle made up of the set of all frames over M. The frame bundle is useful because tensor fields on M can be regarded as equivariant vector-valued functions on F(M).

On a manifold that is sufficiently smooth, various kinds of jet bundles can also be considered. The (first-order) tangent bundle of a manifold is the collection of curves in the manifold modulo the equivalence relation of first-order contact. By analogy, the k-th order tangent bundle is the collection of curves modulo the relation of k-th order contact. Likewise, the cotangent bundle is the bundle of 1-jets of functions on the manifold: the k-jet bundle is the bundle of their k-jets. These and other examples of the general idea of jet bundles play a significant role in the study of differential operators on manifolds.

The notion of a frame also generalizes to the case of higher-order jets. Define a k-th order frame to be the k-jet of a diffeomorphism from R to M. The collection of all k-th order frames, F(M), is a principal G bundle over M, where G is the group of k-jets; i.e., the group made up of k-jets of diffeomorphisms of R that fix the origin. Note that GL(n, R) is naturally isomorphic to G, and a subgroup of every G, k ≥ 2. In particular, a section of F(M) gives the frame components of a connection on M. Thus, the quotient bundle F(M) / GL(n, R) is the bundle of symmetric linear connections over M.