In mathematics, adifferentiable manifold (alsodifferential manifold) is a type ofmanifold that is locally similar enough to avector space to allow one to applycalculus. Anymanifold 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 isdifferentiable), then computations done in one chart are valid in any other differentiable chart.

In formal terms, adifferentiable manifold is atopological manifold with a globally defineddifferential structure. Any topological manifold can be given a differential structure locally by using thehomeomorphisms 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, theircompositions 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 calledtransition 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 differentiabletangent space, differentiable functions, and differentiabletensor andvector fields.

Differentiable manifolds are very important inphysics. Special kinds of differentiable manifolds form the basis for physical theories such asclassical mechanics,general relativity, andYang–Mills theory. It is possible to develop a calculus for differentiable manifolds. This leads to such mathematical machinery as theexterior calculus. The study of calculus on differentiable manifolds is known asdifferential geometry.

"Differentiability" of a manifold has been given several meanings, including:continuously differentiable,k-times differentiable,smooth (which itself has many meanings), andanalytic.

The emergence ofdifferential geometry as a distinct discipline is generally credited toCarl Friedrich Gauss andBernhard Riemann. Riemann first described manifolds in his famoushabilitation lecture before the faculty atGöttingen.^[1] 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 asJames Clerk Maxwell,^[2] and mathematiciansGregorio Ricci-Curbastro andTullio Levi-Civita^[3] led to the development oftensor analysis and the notion ofcovariance, which identifies an intrinsic geometric property as one that is invariant with respect tocoordinate transformations. These ideas found a key application inAlbert Einstein's theory ofgeneral relativity and its underlyingequivalence principle. A modern definition of a 2-dimensional manifold was given byHermann Weyl in his 1913 book onRiemann surfaces.^[4] The widely accepted general definition of a manifold in terms of anatlas is due toHassler Whitney.^[5]

LetM be atopological space. Achart(U,φ) onM consists of an open subsetU ofM, and ahomeomorphismφ fromU to an open subset of someEuclidean spaceRⁿ. Somewhat informally, one may refer to a chartφ :U →Rⁿ, meaning that the image ofφ is an open subset ofRⁿ, and thatφ is a homeomorphism onto its image; in the usage of some authors, this may instead mean thatφ :U →Rⁿ is itself a homeomorphism.

The presence of a chart suggests the possibility of doingdifferential calculus onM; for instance, if given a functionu :M →R and a chart(U,φ) onM, one could consider the compositionu ∘φ⁻¹, 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 itspartial derivatives.

This situation is not fully satisfactory for the following reason. Consider a second chart(V,ψ) onM, and suppose thatU andV contain some points in common. The two corresponding functionsu ∘φ⁻¹ andu ∘ψ⁻¹ are linked in the sense that they can be reparametrized into one another:$${\displaystyle u\circ {\phi }^{-1}=(u\circ {\psi }^{-1})\circ (\psi \circ {\phi }^{-1}),}$$the natural domain of the right-hand side beingφ(U ∩V). Sinceφ andψ are homeomorphisms, it follows thatψ ∘φ⁻¹ is a homeomorphism fromφ(U ∩V) toψ(U ∩V). Consequently, it's just a bicontinuous function, thus even if both functionsu ∘φ⁻¹ andu ∘ψ⁻¹ are differentiable, their differential properties will not necessarily be strongly linked to one another, asψ ∘φ⁻¹ is not guaranteed to be sufficiently differentiable for being able to compute the partial derivatives of the LHS applying thechain rule to the RHS. The same problem is found if one considers instead functionsc :R →M; one is led to the reparametrization formula$${\displaystyle \phi \circ c=(\phi \circ {\psi }^{-1})\circ (\psi \circ c),}$$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 onM for which thetransition mapsψ ∘φ⁻¹ are all differentiable. This makes the situation quite clean: ifu ∘φ⁻¹ is differentiable, then due to the first reparametrization formula listed above, the mapu ∘ψ⁻¹ is also differentiable on the regionψ(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 isM, 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, providedk ≥ 1.

${\displaystyle M}$

${\displaystyle U_{\alpha }}$

${\displaystyle U_{\beta }}$

${\displaystyle {\varphi }_{\alpha }}$

${\displaystyle {\varphi }_{\beta }}$

${\displaystyle {\varphi }_{\beta \alpha }}$

${\displaystyle {\varphi }_{\alpha \beta }}$

${\displaystyle {\mathbf{R}}^n}$

${\displaystyle {\mathbf{R}}^n}$

The transition map of two charts.${\displaystyle {\varphi }_{\alpha \beta }}$ denotes${\displaystyle {\varphi }_{\alpha }\circ {\varphi }_{\beta }^{-1}}$ and${\displaystyle {\varphi }_{\beta \alpha }}$ denotes${\displaystyle {\varphi }_{\beta }\circ {\varphi }_{\alpha }^{-1}}$.

Since every real-analytic map is smooth, and every smooth map isC^k for anyk, one can see that any analytic atlas can also be viewed as a smooth atlas, and every smooth atlas can be viewed as aC^k atlas. This chain can be extended to include holomorphic atlases, with the understanding that any holomorphic map between open subsets ofCⁿ can be viewed as a real-analytic map between open subsets ofR²ⁿ.

Given a differentiable atlas on a topological space, one says that a chart isdifferentiably compatible with the atlas, ordifferentiable 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 amaximal 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 asmooth structure; a maximal holomorphic atlas is also known as acomplex 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 theinvariance of domain, each connected component of a topological space which has a differentiable atlas has a well-defined dimensionn. 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, orC^k atlas. For this reason, one refers separately to the "real" and "complex" dimension of a topological space with a holomorphic atlas.

Adifferentiable manifold is aHausdorff andsecond countable topological spaceM, together with a maximal differentiable atlas onM. 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 ofbump functions andpartitions of unity, both of which are used ubiquitously.

The notion of aC⁰ manifold is identical to that of atopological 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 setM (rather than a topological spaceM), using the natural analogue of a smooth atlas in this setting to define the structure of a topological space onM.

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

A real valued functionf on ann-dimensional differentiable manifoldM is calleddifferentiable at a pointp ∈M if it is differentiable in any coordinate chart defined aroundp. In more precise terms, if${\displaystyle (U,\varphi )}$ is a differentiable chart where${\displaystyle U}$ is an open set in${\displaystyle M}$ containingp and${\displaystyle \varphi :U\to {\mathbf{R}}^n}$ is the map defining the chart, thenf is differentiable atpif and only if$${\displaystyle f\circ {\varphi }^{-1}:\varphi (U)\subset {\mathbf{R}}^n\to \mathbf{R}}$$is differentiable at${\displaystyle \varphi (p)}$, that is${\displaystyle f\circ {\varphi }^{-1}}$ is a differentiable function from the open set${\displaystyle \varphi (U)}$, considered as a subset of${\displaystyle {\mathbf{R}}^n}$, to${\displaystyle \mathbf{R}}$. In general, there will be many available charts; however, the definition of differentiability does not depend on the choice of chart atp. It follows from thechain rule applied to the transition functions between one chart and another that iff is differentiable in any particular chart atp, then it is differentiable in all charts atp. Analogous considerations apply to definingC^k functions, smooth functions, and analytic functions.

There are various ways to define thederivative of a function on a differentiable manifold, the most fundamental of which is thedirectional derivative. The definition of the directional derivative is complicated by the fact that a manifold will lack a suitableaffine structure with which to definevectors. Therefore, the directional derivative looks at curves in the manifold instead of vectors.

Given a real valued functionf on ann dimensional differentiable manifoldM, the directional derivative off at a pointp inM is defined as follows. Suppose that γ(t) is a curve inM withγ(0) =p, which isdifferentiable in the sense that its composition with any chart is adifferentiable curve inRⁿ. Then thedirectional derivative off atp along γ is

$${\displaystyle {\frac{d}{dt}f(\gamma (t))|}_{t=0}.}$$

Ifγ₁ andγ₂ are two curves such thatγ₁(0) =γ₂(0) =p, and in any coordinate chart${\displaystyle \varphi }$,

$${\displaystyle {\frac{d}{dt}\varphi \circ {\gamma }_1(t)|}_{t=0}={\frac{d}{dt}\varphi \circ {\gamma }_2(t)|}_{t=0}}$$

then, by the chain rule,f has the same directional derivative atp alongγ₁ as alongγ₂. This means that the directional derivative depends only on thetangent vector of the curve atp. 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.

Atangent vector atp ∈M is anequivalence class of differentiable curvesγ withγ(0) =p, modulo the equivalence relation of first-ordercontact between the curves. Therefore,

$${\displaystyle {\gamma }_1\equiv {\gamma }_2⟺{\frac{d}{dt}\varphi \circ {\gamma }_1(t)|}_{t=0}={\frac{d}{dt}\varphi \circ {\gamma }_2(t)|}_{t=0}}$$

in every coordinate chart${\displaystyle \varphi }$. Therefore, the equivalence classes are curves throughp with a prescribedvelocity vector atp. The collection of all tangent vectors atp forms avector space: thetangent space toM atp, denotedT_pM.

IfX is a tangent vector atp andf a differentiable function defined nearp, then differentiatingf along any curve in the equivalence class definingX gives a well-defined directional derivative alongX:$${\displaystyle Xf(p):={\frac{d}{dt}f(\gamma (t))|}_{t=0}.}$$Once again, the chain rule establishes that this is independent of the freedom in selecting γ from the equivalence class, since any curve with the same first order contact will yield the same directional derivative.

If the functionf is fixed, then the mapping$${\displaystyle X\mapsto Xf(p)}$$is alinear functional on the tangent space. This linear functional is often denoted bydf(p) and is called thedifferential off atp:$${\displaystyle df(p):T_{p}M\to \mathbf{R}.}$$

Let${\displaystyle M}$ be a topological${\displaystyle n}$-manifold with a smooth atlas${\displaystyle \{(U_{\alpha },{\varphi }_{\alpha }){\}}_{\alpha \in A}.}$ Given${\displaystyle p\in M}$ let${\displaystyle A_p}$ denote${\displaystyle \{\alpha \in A:p\in U_{\alpha }\}.}$ A "tangent vector at${\displaystyle p\in M}$" is a mapping${\displaystyle v:A_p\to {\mathbb{R}}^n,}$ here denoted${\displaystyle \alpha \mapsto v_{\alpha },}$ such that$${\displaystyle v_{\alpha }=D{|}_{{\varphi }_{\beta }(p)}({\varphi }_{\alpha }\circ {\varphi }_{\beta }^{-1})(v_{\beta })}$$for all${\displaystyle \alpha ,\beta \in A_p.}$ Let the collection of tangent vectors at${\displaystyle p}$ be denoted by${\displaystyle T_{p}M.}$ Given a smooth function${\displaystyle f:M\to \mathbb{R}}$, define${\displaystyle d{f}_p:T_{p}M\to \mathbb{R}}$ by sending a tangent vector${\displaystyle v:A_p\to {\mathbb{R}}^n}$ to the number given by$${\displaystyle D{|}_{{\varphi }_{\alpha }(p)}(f\circ {\varphi }_{\alpha }^{-1})(v_{\alpha }),}$$which due to the chain rule and the constraint in the definition of a tangent vector does not depend on the choice of${\displaystyle \alpha \in A_p.}$

One can check that${\displaystyle T_{p}M}$ naturally has the structure of a${\displaystyle n}$-dimensional real vector space, and that with this structure,${\displaystyle d{f}_p}$ is a linear map. The key observation is that, due to the constraint appearing in the definition of a tangent vector, the value of${\displaystyle v_{\beta }}$ for a single element${\displaystyle \beta }$ of${\displaystyle A_p}$ automatically determines${\displaystyle v_{\alpha }}$ for all${\displaystyle \alpha \in A.}$

The above formal definitions correspond precisely to a more informal notation which appears often in textbooks, specifically

${\displaystyle v^i={\tilde{v}}^j\frac{\mathrm{\partial }{x}^i}{\mathrm{\partial }{\tilde{x}}^j}}$ and${\displaystyle d{f}_p(v)=\frac{\mathrm{\partial }f}{\mathrm{\partial }{x}^i}{v}^i.}$

With the idea of the formal definitions understood, this shorthand notation is, for most purposes, much easier to work with.

One of the topological features of the sheaf of differentiable functions on a differentiable manifold is that it admitspartitions 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 thatM is a manifold of classC^k, where0 ≤k ≤ ∞. Let {U_α} be an open covering ofM. Then apartition of unity subordinate to the cover {U_α} is a collection of real-valuedC^k functionsφ_i onM satisfying the following conditions:

• Thesupports of theφ_i arecompact andlocally finite;
• The support ofφ_i is completely contained inU_α for someα;
• Theφ_i sum to one at each point ofM:$${\displaystyle \sum _i{\varphi }_i(x)=1.}$$

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

Every open covering of aC^k manifoldM has aC^k partition of unity. This allows for certain constructions from the topology ofC^k functions onRⁿ 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 ofRⁿ. Partitions of unity therefore allow for certain other kinds offunction spaces to be considered: for instanceL^p spaces,Sobolev spaces, and other kinds of spaces that require integration.

SupposeM andN are two differentiable manifolds with dimensionsm andn, respectively, andf is a function fromM toN. Since differentiable manifolds are topological spaces we know what it means forf to be continuous. But what does "f isC^k(M,N)" mean fork ≥ 1? We know what that means whenf is a function between Euclidean spaces, so if we composef with a chart ofM and a chart ofN such that we get a map that goes from Euclidean space toM toN to Euclidean space we know what it means for that map to beC^k(R^m,Rⁿ). We define "f isC^k(M,N)" to mean that all such compositions off with charts areC^k(R^m,Rⁿ). Once again, the chain rule guarantees that the idea of differentiability does not depend on which charts of the atlases onM andN are selected. However, defining the derivative itself is more subtle. IfM orN is itself already a Euclidean space, then we don't need a chart to map it to one.

Thetangent space of a point consists of the possible directional derivatives at that point, and has the samedimensionn as does the manifold. For a set of (non-singular) coordinatesx_k local to the point, the coordinate derivatives${\displaystyle {\mathrm{\partial }}_k=\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_k}}$ define aholonomic basis of the tangent space. The collection of tangent spaces at all points can in turn be made into a manifold, thetangent bundle, whose dimension is 2n. The tangent bundle is wheretangent vectors lie, and is itself a differentiable manifold. TheLagrangian is a function on the tangent bundle. One can also define the tangent bundle as the bundle of 1-jets fromR (thereal line) toM.

One may construct an atlas for the tangent bundle consisting of charts based onU_α ×Rⁿ, whereU_α denotes one of the charts in the atlas forM. Each of these new charts is the tangent bundle for the chartsU_α. The transition maps on this atlas are defined from the transition maps on the original manifold, and retain the original differentiability class.

Thedual space of a vector space is the set of real valued linear functions on the vector space. Thecotangent space at a point is the dual of the tangent space at that point and the elements are referred to as cotangent vectors; thecotangent 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. TheHamiltonian is a scalar on the cotangent bundle. Thetotal space of a cotangent bundle has the structure of asymplectic manifold. Cotangent vectors are sometimes calledcovectors. One can also define the cotangent bundle as the bundle of 1-jets of functions fromM toR.

Elements of the cotangent space can be thought of asinfinitesimal displacements: iff is a differentiable function we can define at each pointp a cotangent vectordf_p, which sends a tangent vectorX_p to the derivative off associated withX_p. However, not every covector field can be expressed this way. Those that can are referred to asexact differentials. For a given set of local coordinatesx^k, the differentialsdx^k
_p form a basis of the cotangent space atp.

The tensor bundle is thedirect sum of alltensor products of the tangent bundle and the cotangent bundle. Each element of the bundle is atensor field, which can act as amultilinear 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 analgebra 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 ascovariant andcontravariant 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 ofRⁿ 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), aGL(n,R)principal bundle made up of the set of all frames overM. The frame bundle is useful because tensor fields onM can be regarded asequivariant 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-ordercontact. By analogy, thek-th order tangent bundle is the collection of curves modulo the relation ofk-th order contact. Likewise, the cotangent bundle is the bundle of 1-jets of functions on the manifold: thek-jet bundle is the bundle of theirk-jets. These and other examples of the general idea of jet bundles play a significant role in the study ofdifferential operators on manifolds.

The notion of a frame also generalizes to the case of higher-order jets. Define ak-th order frame to be thek-jet of adiffeomorphism fromRⁿ toM.^[6] The collection of allk-th order frames,F^k(M), is a principalG^k bundle overM, whereG^k is thegroup ofk-jets; i.e., the group made up ofk-jets of diffeomorphisms ofRⁿ that fix the origin. Note thatGL(n,R) is naturally isomorphic toG¹, and a subgroup of everyG^k,k ≥ 2. In particular, a section ofF²(M) gives the frame components of aconnection onM. Thus, the quotient bundleF²(M) / GL(n,R) is the bundle ofsymmetric linear connections overM.

Many of the techniques frommultivariate calculus also apply,mutatis mutandis, to differentiable manifolds. One can define the directional derivative of a differentiable function along a tangent vector to the manifold, for instance, and this leads to a means of generalizing thetotal derivative of a function: the differential. From the perspective of calculus, the derivative of a function on a manifold behaves in much the same way as the ordinary derivative of a function defined on a Euclidean space, at leastlocally. For example, there are versions of theimplicit andinverse function theorems for such functions.

There are, however, important differences in the calculus of vector fields (and tensor fields in general). In brief, the directional derivative of a vector field is not well-defined, or at least not defined in a straightforward manner. Several generalizations of the derivative of a vector field (or tensor field) do exist, and capture certain formal features of differentiation in Euclidean spaces. The chief among these are:

• TheLie derivative, which is uniquely defined by the differential structure, but fails to satisfy some of the usual features of directional differentiation.
• Anaffine connection, which is not uniquely defined, but generalizes in a more complete manner the features of ordinary directional differentiation. Because an affine connection is not unique, it is an additional piece of data that must be specified on the manifold.

Ideas fromintegral calculus also carry over to differential manifolds. These are naturally expressed in the language ofexterior calculus anddifferential forms. The fundamental theorems of integral calculus in several variables—namelyGreen's theorem, thedivergence theorem, andStokes' theorem—generalize to a theorem (also called Stokes' theorem) relating theexterior derivative and integration oversubmanifolds.

Differentiable functions between two manifolds are needed in order to formulate suitable notions ofsubmanifolds, and other related concepts. Iff :M →N is a differentiable function from a differentiable manifoldM of dimensionm to another differentiable manifoldN of dimensionn, then thedifferential off is a mappingdf : TM → TN. It is also denoted byTf and called thetangent map. At each point ofM, this is a linear transformation from one tangent space to another:$${\displaystyle df(p):T_{p}M\to T_{f(p)}N.}$$Therank off atp is therank of this linear transformation.

Usually the rank of a function is a pointwise property. However, if the function has maximal rank, then the rank will remain constant in a neighborhood of a point. A differentiable function "usually" has maximal rank, in a precise sense given bySard's theorem. Functions of maximal rank at a point are calledimmersions andsubmersions:

• Ifm ≤n, andf :M →N has rankm atp ∈M, thenf is called animmersion atp. Iff is an immersion at all points ofM and is ahomeomorphism onto its image, thenf is anembedding. Embeddings formalize the notion ofM being asubmanifold ofN. In general, an embedding is an immersion without self-intersections and other sorts of non-local topological irregularities.
• Ifm ≥n, andf :M →N has rankn atp ∈M, thenf is called asubmersion atp. The implicit function theorem states that iff is a submersion atp, thenM is locally a product ofN andR^m−n nearp. In formal terms, there exist coordinates(y₁, ...,y_n) in a neighborhood off(p) inN, andm −n functionsx₁, ...,x_m−n defined in a neighborhood ofp inM such that$${\displaystyle (y_1\circ f,\dots ,y_n\circ f,x_1,\dots ,x_{m-n})}$$ is a system of local coordinates ofM in a neighborhood ofp. Submersions form the foundation of the theory offibrations andfibre bundles.

ALie derivative, named afterSophus Lie, is aderivation on thealgebra oftensor fields over amanifoldM. Thevector space of all Lie derivatives onM forms an infinite dimensionalLie algebra with respect to theLie bracket defined by

$${\displaystyle [A,B]:={\mathcal{L}}_{A}B=-{\mathcal{L}}_{B}A.}$$

The Lie derivatives are represented byvector fields, asinfinitesimal generators of flows (activediffeomorphisms) onM. Looking at it the other way around, thegroup of diffeomorphisms ofM has the associated Lie algebra structure, of Lie derivatives, in a way directly analogous to theLie group theory.

The exterior calculus allows for a generalization of thegradient,divergence andcurl operators.

The bundle ofdifferential forms, at each point, consists of all totallyantisymmetricmultilinear maps on the tangent space at that point. It is naturally divided inton-forms for eachn at most equal to the dimension of the manifold; ann-form is ann-variable form, also called a form of degreen. The 1-forms are the cotangent vectors, while the 0-forms are just scalar functions. In general, ann-form is a tensor with cotangent rankn and tangent rank 0. But not every such tensor is a form, as a form must be antisymmetric.

Theexterior derivative is a linear operator on thegraded vector space of all smooth differential forms on a smooth manifold${\displaystyle M}$. It is usually denoted by${\displaystyle d}$. More precisely, if${\displaystyle n=\dim (M)}$, for${\displaystyle 0\le k\le n}$ the operator${\displaystyle d}$ maps the space${\displaystyle {\mathrm{\Omega }}^k(M)}$ of${\displaystyle k}$-forms on${\displaystyle M}$ into the space${\displaystyle {\mathrm{\Omega }}^{k+1}(M)}$ of${\displaystyle (k+1)}$-forms (if${\displaystyle k>n}$ there are no non-zero${\displaystyle k}$-forms on${\displaystyle M}$ so the map${\displaystyle d}$ is identically zero on${\displaystyle n}$-forms).

For example, the exterior differential of a smooth function${\displaystyle f}$ is given in local coordinates${\displaystyle x_1,\dots ,x_n}$, with associated local co-frame${\displaystyle d{x}_1,\dots ,d{x}_n}$ by the formula :$${\displaystyle df=\sum _{i=1}^n\frac{\mathrm{\partial }f}{\mathrm{\partial }{x}_i}d{x}_i.}$$

The exterior differential satisfies the following identity, similar to aproduct rule with respect to the wedge product of forms:$${\displaystyle d(\omega \wedge \eta )=d\omega \wedge \eta +(-1{)}^{\mathrm{deg}\omega }\omega \wedge d\eta .}$$

The exterior derivative also satisfies the identity${\displaystyle d\circ d=0}$. That is, if${\displaystyle \omega }$ is a${\displaystyle k}$-form then the${\displaystyle (k+2)}$-form${\displaystyle d(df)}$ is identically vanishing. A form${\displaystyle \omega }$ such that${\displaystyle d\omega =0}$ is calledclosed, while a form${\displaystyle \omega }$ such that${\displaystyle \omega =d\eta }$ for some other form${\displaystyle \eta }$ is calledexact. Another formulation of the identity${\displaystyle d\circ d=0}$ is that an exact form is closed. This allows one to definede Rham cohomology of the manifold${\displaystyle M}$, where the${\displaystyle k}$th cohomology group is thequotient group of the closed forms on${\displaystyle M}$ by the exact forms on${\displaystyle M}$.

Suppose that${\displaystyle M}$ is a topological${\displaystyle n}$-manifold.

If given any smooth atlas${\displaystyle \{(U_{\alpha },{\varphi }_{\alpha }){\}}_{\alpha \in A}}$, it is easy to find a smooth atlas which defines a different smooth manifold structure on${\displaystyle M;}$ consider a homeomorphism${\displaystyle \mathrm{\Phi }:M\to M}$ which is not smooth relative to the given atlas; for instance, one can modify the identity map using a localized non-smooth bump. Then consider the new atlas${\displaystyle \{({\mathrm{\Phi }}^{-1}(U_{\alpha }),{\varphi }_{\alpha }\circ \mathrm{\Phi }){\}}_{\alpha \in A},}$ which is easily verified as a smooth atlas. However, the charts in the new atlas are not smoothly compatible with the charts in the old atlas, since this would require that${\displaystyle {\varphi }_{\alpha }\circ \mathrm{\Phi }\circ {\varphi }_{\beta }^{-1}}$ and${\displaystyle {\varphi }_{\alpha }\circ {\mathrm{\Phi }}^{-1}\circ {\varphi }_{\beta }^{-1}}$ are smooth for any${\displaystyle \alpha }$ and${\displaystyle \beta ,}$ with these conditions being exactly the definition that both${\displaystyle \mathrm{\Phi }}$ and${\displaystyle {\mathrm{\Phi }}^{-1}}$ are smooth, in contradiction to how${\displaystyle \mathrm{\Phi }}$ was selected.

With this observation as motivation, one can define an equivalence relation on the space of smooth atlases on${\displaystyle M}$ by declaring that smooth atlases${\displaystyle \{(U_{\alpha },{\varphi }_{\alpha }){\}}_{\alpha \in A}}$ and${\displaystyle \{(V_{\beta },{\psi }_{\beta }){\}}_{\beta \in B}}$ are equivalent if there is a homeomorphism${\displaystyle \mathrm{\Phi }:M\to M}$ such that${\displaystyle \{({\mathrm{\Phi }}^{-1}(U_{\alpha }),{\varphi }_{\alpha }\circ \mathrm{\Phi }){\}}_{\alpha \in A}}$ is smoothly compatible with${\displaystyle \{(V_{\beta },{\psi }_{\beta }){\}}_{\beta \in B},}$ and such that${\displaystyle \{(\mathrm{\Phi }(V_{\beta }),{\psi }_{\beta }\circ {\mathrm{\Phi }}^{-1}){\}}_{\beta \in B}}$ is smoothly compatible with${\displaystyle \{(U_{\alpha },{\varphi }_{\alpha }){\}}_{\alpha \in A}.}$

More briefly, one could say that two smooth atlases are equivalent if there exists a diffeomorphism${\displaystyle M\to M,}$ in which one smooth atlas is taken for the domain and the other smooth atlas is taken for the range.

Note that this equivalence relation is a refinement of the equivalence relation which defines a smooth manifold structure, as any two smoothly compatible atlases are also compatible in the present sense; one can take${\displaystyle \mathrm{\Phi }}$ to be the identity map.

If the dimension of${\displaystyle M}$ is 1, 2, or 3, then there exists a smooth structure on${\displaystyle M}$, and all distinct smooth structures are equivalent in the above sense. The situation is more complicated in higher dimensions, although it isn't fully understood.

• Some topological manifolds admit no smooth structures, as was originally shown with aten-dimensional example byKervaire (1960). Amajor application ofpartial differential equations in differential geometry due toSimon Donaldson, in combination with results ofMichael Freedman, shows that many simply-connected compact topological 4-manifolds do not admit smooth structures. A well-known particular example is theE₈ manifold.
• Some topological manifolds admit many smooth structures which are not equivalent in the sense given above. This was originally discovered byJohn Milnor in the form of theexotic 7-spheres.^[7]

Every one-dimensional connected smooth manifold is diffeomorphic to either${\displaystyle \mathbb{R}}$ or${\displaystyle S^1,}$ each with their standard smooth structures.

For a classification of smooth 2-manifolds, seesurface. A particular result is that every two-dimensional connected compact smooth manifold is diffeomorphic to one of the following:${\displaystyle S^2,}$ or${\displaystyle (S^1\times S^1)\mathrm{♯}\cdots \mathrm{♯}(S^1\times S^1),}$ or${\displaystyle {\mathbb{R}\mathbb{P}}^2\mathrm{♯}\cdots \mathrm{♯}{\mathbb{R}\mathbb{P}}^2.}$ The situation ismore nontrivial if one considers complex-differentiable structure instead of smooth structure.

The situation in three dimensions is quite a bit more complicated, and known results are more indirect. A remarkable result, proved in 2002 by methods ofpartial differential equations, is thegeometrization conjecture, stating loosely that any compact smooth 3-manifold can be split up into different parts, each of which admits Riemannian metrics which possess many symmetries. There are also various "recognition results" for geometrizable 3-manifolds, such asMostow rigidity and Sela's algorithm for the isomorphism problem for hyperbolic groups.^[8]

The classification ofn-manifolds forn greater than three is known to be impossible, even up tohomotopy equivalence. Given any finitelypresented group, one can construct a closed 4-manifold having that group as fundamental group. Since there is no algorithm todecide the isomorphism problem for finitely presented groups, there is no algorithm to decide whether two 4-manifolds have the same fundamental group. Since the previously described construction results in a class of 4-manifolds that are homeomorphic if and only if their groups are isomorphic, the homeomorphism problem for 4-manifolds isundecidable. In addition, since even recognizing the trivial group is undecidable, it is not even possible in general to decide whether a manifold has trivial fundamental group, i.e. issimply connected.

Simply connected4-manifolds have been classified up to homeomorphism byFreedman using theintersection form andKirby–Siebenmann invariant. Smooth 4-manifold theory is known to be much more complicated, as theexotic smooth structures onR⁴ demonstrate.

However, the situation becomes more tractable for simply connected smooth manifolds of dimension ≥ 5, where theh-cobordism theorem can be used to reduce the classification to a classification up to homotopy equivalence, andsurgery theory can be applied.^[9] This has been carried out to provide an explicit classification of simply connected5-manifolds by Dennis Barden.

ARiemannian manifold consists of a smooth manifold together with a positive-definiteinner product on each of the individual tangent spaces. This collection of inner products is called theRiemannian metric, and is naturally a symmetric 2-tensor field. This "metric" identifies a natural vector space isomorphism${\displaystyle T_{p}M\to T_p^{\ast }M}$ for each${\displaystyle p\in M.}$ On a Riemannian manifold one can define notions of length, volume, and angle. Any smooth manifold can be given many different Riemannian metrics.

Apseudo-Riemannian manifold (also called a semi-Riemannian manifold) is a generalization of the notion ofRiemannian manifold where the inner products are allowed to have anindefinite signature, as opposed to beingpositive-definite; they are still required to be non-degenerate. Every smooth pseudo-Riemannian and Riemmannian manifold defines a number of associated tensor fields, such as theRiemann curvature tensor.Lorentzian manifolds are pseudo-Riemannian manifolds of signature${\displaystyle (n-1,1)}$; the case${\displaystyle n=4}$ is fundamental ingeneral relativity. Not every smooth manifold can be given a non-Riemannian pseudo-Riemannian structure; there are topological restrictions on doing so.

AFinsler manifold is a different generalization of a Riemannian manifold, in which the inner product is replaced with avector norm; as such, this allows the definition of length, but not angle.

Asymplectic manifold is a manifold equipped with aclosed,nondegenerate2-form. This condition forces symplectic manifolds to be even-dimensional, due to the fact that skew-symmetric${\displaystyle (2n+1)\times (2n+1)}$ matrices all have zero determinant. There are two basic examples:

• Cotangent bundles, which arise as phase spaces inHamiltonian mechanics, are a motivating example, since they admit anatural symplectic form.
• All oriented two-dimensional Riemannian manifolds${\displaystyle (M,g)}$ are, in a natural way, symplectic, by defining the form${\displaystyle \omega (u,v)=g(u,J(v))}$ where, for any${\displaystyle v\in T_{p}M,}$${\displaystyle J(v)}$ denotes the vector such that${\displaystyle v,J(v)}$ is an oriented${\displaystyle g_p}$-orthonormal basis of${\displaystyle T_{p}M.}$

ALie group consists of aC^∞ manifold${\displaystyle G}$ together with agroup structure on${\displaystyle G}$ such that the product and inversion maps${\displaystyle m:G\times G\to G}$ and${\displaystyle i:G\to G}$ are smooth as maps of manifolds. These objects often arise naturally in describing (continuous) symmetries, and they form an important source of examples of smooth manifolds.

Many otherwise familiar examples of smooth manifolds, however, cannot be given a Lie group structure, since given a Lie group${\displaystyle G}$ and any${\displaystyle g\in G}$, one could consider the map${\displaystyle m(g,\cdot ):G\to G}$ which sends the identity element${\displaystyle e}$ to${\displaystyle g}$ and hence, by considering the differential${\displaystyle T_{e}G\to T_{g}G,}$ gives a natural identification between any two tangent spaces of a Lie group. In particular, by considering an arbitrary nonzero vector in${\displaystyle T_{e}G,}$ one can use these identifications to give a smooth non-vanishing vector field on${\displaystyle G.}$ This shows, for instance, that noeven-dimensional sphere can support a Lie group structure. The same argument shows, more generally, that every Lie group must beparallelizable.

The notion of apseudogroup^[10] provides a flexible generalization of atlases in order to allow a variety of different structures to be defined on manifolds in a uniform way. Apseudogroup consists of a topological spaceS and a collection Γ consisting of homeomorphisms from open subsets ofS to other open subsets ofS such that

1. Iff ∈ Γ, andU is an open subset of the domain off, then the restrictionf|_U is also in Γ.
2. Iff is a homeomorphism from a union of open subsets ofS,${\displaystyle {\cup }_i{U}_i}$, to an open subset ofS, thenf ∈ Γ provided${\displaystyle f{|}_{U_i}\in \mathrm{\Gamma }}$ for everyi.
3. For every openU ⊂S, the identity transformation ofU is in Γ.
4. Iff ∈ Γ, thenf⁻¹ ∈ Γ.
5. The composition of two elements of Γ is in Γ.

These last three conditions are analogous to the definition of agroup. Note that Γ need not be a group, however, since the functions are not globally defined onS. For example, the collection of all localC^kdiffeomorphisms onRⁿ form a pseudogroup. Allbiholomorphisms between open sets inCⁿ form a pseudogroup. More examples include: orientation preserving maps ofRⁿ,symplectomorphisms,Möbius transformations,affine transformations, and so on. Thus, a wide variety of function classes determine pseudogroups.

An atlas (U_i,φ_i) of homeomorphismsφ_i fromU_i ⊂M to open subsets of a topological spaceS is said to becompatible with a pseudogroup Γ provided that the transition functionsφ_j ∘φ_i⁻¹ :φ_i(U_i ∩U_j) →φ_j(U_i ∩U_j) are all in Γ.

A differentiable manifold is then an atlas compatible with the pseudogroup ofC^k functions onRⁿ. A complex manifold is an atlas compatible with the biholomorphic functions on open sets inCⁿ. 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 aC^k-structure. Herek = 1, 2, ..., ∞, or ω for real analytic manifolds. Instead of considering coordinate charts, it is possible to start with functions defined on the manifold itself. Thestructure sheaf ofM, denotedC^k, is a sort offunctor that defines, for each open setU ⊂M, an algebraC^k(U) of continuous functionsU →R. A structure sheafC^k is said to giveM the structure of aC^k manifold of dimensionn provided that, for anyp ∈M, there exists a neighborhoodU ofp andn functionsx¹, ...,xⁿ ∈C^k(U) such that the mapf = (x¹, ...,xⁿ) :U →Rⁿ is a homeomorphism onto an open set inRⁿ, and such thatC^k|_U is thepullback of the sheaf ofk-times continuously differentiable functions onRⁿ.^[11]

In particular, this latter condition means that any functionh inC^k(V), forV, can be written uniquely ash(x) =H(x¹(x), ...,xⁿ(x)), whereH is ak-times differentiable function onf(V) (an open set inRⁿ). Thus, the sheaf-theoretic viewpoint is that the functions on a differentiable manifold can be expressed in local coordinates as differentiable functions onRⁿ, anda 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 aringed space. This approach is strongly influenced by the theory ofschemes inalgebraic geometry, but useslocal rings of thegerms of differentiable functions. It is especially popular in the context ofcomplex manifolds.

We begin by describing the basic structure sheaf onRⁿ. IfU is an open set inRⁿ, let

O(U) =C^k(U,R)

consist of all real-valuedk-times continuously differentiable functions onU. AsU varies, this determines a sheaf of rings onRⁿ. The stalkO_p forp ∈Rⁿ consists ofgerms of functions nearp, and is an algebra overR. In particular, this is alocal ring whose uniquemaximal ideal consists of those functions that vanish atp. The pair(Rⁿ,O) is an example of alocally ringed space: it is a topological space equipped with a sheaf whose stalks are each local rings.

A differentiable manifold (of classC^k) consists of a pair(M,O_M) whereM is asecond countableHausdorff space, andO_M is a sheaf of localR-algebras defined onM, such that the locally ringed space(M,O_M) is locally isomorphic to(Rⁿ,O). In this way, differentiable manifolds can be thought of asschemes modeled onRⁿ. This means that^[12] for each pointp ∈M, there is a neighborhoodU ofp, and a pair of functions(f,f^#), where

1. f :U →f(U) ⊂Rⁿ is a homeomorphism onto an open set inRⁿ.
2. f^#:O|_f(U) →f_∗ (O_M|_U) is an isomorphism of sheaves.
3. The localization off^# is an isomorphism of local rings

f^#_f(p) :O_f(p) →O_M,p.

There are a number of important motivations for studying differentiable manifolds within this abstract framework. First, there is noa priori reason that the model space needs to beRⁿ. For example, (in particular inalgebraic geometry), one could take this to be the space of complex numbersCⁿ equipped with the sheaf ofholomorphic functions (thus arriving at the spaces ofcomplex analytic geometry), or the sheaf ofpolynomials (thus arriving at the spaces of interest in complexalgebraic geometry). In broader terms, this concept can be adapted for any suitable notion of a scheme (seetopos 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 oflocal isomorphism rather than being central to the discussion (as in the case of charts and atlases). Third, the sheafO_M is not manifestly a sheaf of functions at all. Rather, it emerges as a sheaf of functions as aconsequence of the construction (via the quotients of local rings by their maximal ideals). Hence, it is a more primitive definition of the structure (seesynthetic 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.

• Thecotangent space at a point isI_p/I_p², whereI_p is the maximal ideal of the stalkO_M,p.
• In general, the entirecotangent bundle can be obtained by a related technique (seecotangent bundle for details).
• Taylor series (andjets) can be approached in a coordinate-independent manner using theI_p-adic filtration onO_M,p.
• Thetangent bundle (or more precisely its sheaf of sections) can be identified with the sheaf of morphisms ofO_M into the ring ofdual numbers.

Thecategory of smooth manifolds with smooth maps lacks certain desirable properties, and people have tried to generalize smooth manifolds in order to rectify this.Diffeological spaces use a different notion of chart known as a "plot".Frölicher spaces andorbifolds are other attempts.

Arectifiable set generalizes the idea of a piece-wise smooth orrectifiable curve to higher dimensions; however, rectifiable sets are not in general manifolds.

Banach manifolds andFréchet manifolds, in particularmanifolds of mappingsare infinite dimensional differentiable manifolds.

This sectionneeds expansion. You can help byadding to it.(June 2008)

For aC^k manifoldM, theset of real-valuedC^k functions on the manifold forms analgebra under pointwise addition and multiplication, called thealgebra of scalar fields or simply thealgebra of scalars. This algebra has the constant function 1 as the multiplicative identity, and is a differentiable analog of the ring ofregular functions in algebraic geometry.

It is possible to reconstruct a manifold from its algebra of scalars, first as a set, but also as a topological space – this is an application of theBanach–Stone theorem, and is more formally known as thespectrum of a C*-algebra. First, there is a one-to-one correspondence between the points ofM and the algebra homomorphismsφ:C^k(M) →R, as such a homomorphismφ corresponds to a codimension one ideal inC^k(M) (namely the kernel ofφ), which is necessarily a maximal ideal. 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) ofC^k(M) recoversM as a point set, though in fact it recoversM 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) andoperator theory (interpreting Banach spaces geometrically). For example, the tangent bundle toM can be defined as the derivations of the algebra of smooth functions onM.

This "algebraization" of a manifold (replacing a geometric object with an algebra) leads to the notion of aC*-algebra – a commutative C*-algebra being precisely the ring of scalars of a manifold, by Banach–Stone, and allows one to considernoncommutative C*-algebras as non-commutative generalizations of manifolds. This is the basis of the field ofnoncommutative geometry.

• Smooth manifolds

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