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The purpose of these notes is to introduce and study differentiable mani- folds. Differentiable manifolds are the central objects in differential geometry, and they .

**Table of contents**

- Differentiable manifold - Encyclopedia of Mathematics
- Differentiable manifolds 2 (BM)
- A Workbook for Students and Teachers

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## Differentiable manifold - Encyclopedia of Mathematics

Line and surface integrals Divergence and curl of vector fields. 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.

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.

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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.

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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. 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.

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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 n to M. In particular, a section of F 2 M gives the frame components of a connection on M. Many of the techniques from multivariate 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 the total 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 least locally.

For example, there are versions of the implicit and inverse 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:. Ideas from integral calculus also carry over to differential manifolds. These are naturally expressed in the language of exterior calculus and differential forms. The fundamental theorems of integral calculus in several variables—namely Green's theorem , the divergence theorem , and Stokes' theorem —generalize to a theorem also called Stokes' theorem relating the exterior derivative and integration over submanifolds. Differentiable functions between two manifolds are needed in order to formulate suitable notions of submanifolds , and other related concepts. It is also denoted by Tf and called the tangent map.

At each point of M , this is a linear transformation from one tangent space to another:. The rank of f at p is the rank 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 by Sard's theorem. Functions of maximal rank at a point are called immersions and submersions :. A Lie derivative , named after Sophus Lie , is a derivation on the algebra of tensor fields over a manifold M. The vector space of all Lie derivatives on M forms an infinite dimensional Lie algebra with respect to the Lie bracket defined by.

The Lie derivatives are represented by vector fields , as infinitesimal generators of flows active diffeomorphisms on M. Looking at it the other way around, the group of diffeomorphisms of M has the associated Lie algebra structure, of Lie derivatives, in a way directly analogous to the Lie group theory. The exterior calculus allows for a generalization of the gradient , divergence and curl operators. The bundle of differential forms , at each point, consists of all totally antisymmetric multilinear maps on the tangent space at that point.

It is naturally divided into n -forms for each n at most equal to the dimension of the manifold; an n -form is an n -variable form, also called a form of degree n. The 1-forms are the cotangent vectors, while the 0-forms are just scalar functions. In general, an n -form is a tensor with cotangent rank n and tangent rank 0. But not every such tensor is a form, as a form must be antisymmetric. There is a map from scalars to covectors called the exterior derivative. This map is the one that relates covectors to infinitesimal displacements, mentioned above; some covectors are the exterior derivatives of scalar functions.

Applying this derivative twice will produce a zero form. Forms with zero derivative are called closed forms, while forms that are themselves exterior derivatives are known as exact forms. The exterior derivative extends to this algebra, and satisfies a version of the product rule :. From the differential forms and the exterior derivative, one can define the de Rham cohomology of the manifold. The rank n cohomology group is the quotient group of the closed forms by the exact forms.

Every topological manifold in dimension 1, 2, or 3 has a unique differential structure up to diffeomorphism ; thus the concepts of topological and differentiable manifold are distinct only in higher dimensions. It is known that in each higher dimension, there are some topological manifolds with no smooth structure, and some with multiple non-diffeomorphic structures.

The existence of non-smoothable manifolds was proven by Kervaire , see Kervaire manifold , and later explained in the context of Donaldson's theorem compare Hilbert's fifth problem ; [11] a good example of a non-smoothable manifold is the E 8 manifold. The classic example of manifolds with multiple incompatible structures are the exotic 7-spheres of John Milnor. Every second-countable 1-manifold without boundary is homeomorphic to a disjoint union of countably many copies of R the real line and S the circle ; the only connected examples are R and S , and of these only S is compact.

In higher dimensions, classification theory normally focuses only on compact connected manifolds. For a classification of 2-manifolds, see surface : in particular compact connected oriented 2-manifolds are classified by their genus, which is a nonnegative integer.

## Differentiable manifolds 2 (BM)

A classification of 3-manifolds follows in principle from the geometrization of 3-manifolds and various recognition results for geometrizable 3-manifolds, such as Mostow rigidity and Sela's algorithm for the isomorphism problem for hyperbolic groups. The classification of n -manifolds for n greater than three is known to be impossible, even up to homotopy equivalence. Given any finitely presented group, one can construct a closed 4-manifold having that group as fundamental group. Since there is no algorithm to decide 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 is undecidable. 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. Simply connected 4-manifolds have been classified up to homeomorphism by Freedman using the intersection form and Kirby—Siebenmann invariant.

Smooth 4-manifold theory is known to be much more complicated, as the exotic smooth structures on R 4 demonstrate. A Riemannian manifold is a differentiable manifold on which the tangent spaces are equipped with inner products in a differentiable fashion. The inner product structure is given in the form of a symmetric 2-tensor called the Riemannian metric. This metric can be used to interconvert vectors and covectors, and to define a rank 4 Riemann curvature tensor.

On a Riemannian manifold one has notions of length, volume, and angle. Any differentiable manifold can be given a Riemannian structure. A pseudo-Riemannian manifold is a variant of Riemannian manifold where the metric tensor is allowed to have an indefinite signature as opposed to a positive-definite one. Pseudo-Riemannian manifolds of signature 3, 1 are important in general relativity. Not every differentiable manifold can be given a strictly pseudo-Riemannian structure; there are topological restrictions on doing so.

A Finsler manifold is a generalization of a Riemannian manifold, in which the inner product is replaced with a vector norm ; this allows the definition of length, but not angle. A symplectic manifold is a manifold equipped with a closed , nondegenerate 2-form. This condition forces symplectic manifolds to be even-dimensional.

Cotangent bundles, which arise as phase spaces in Hamiltonian mechanics , are the motivating example, but many compact manifolds also have symplectic structure. All orientable surfaces embedded in Euclidean space have a symplectic structure , the signed area form on each tangent space induced by the ambient Euclidean inner product. These objects arise naturally in describing symmetries.

The category 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". A rectifiable set generalizes the idea of a piece-wise smooth or rectifiable curve to higher dimensions; however, rectifiable sets are not in general manifolds. From Wikipedia, the free encyclopedia.

Manifold upon which it is possible to perform calculus. A nondifferentiable atlas of charts for the globe. The results of calculus may not be compatible between charts if the atlas is not differentiable. In the center and right charts, the Tropic of Cancer is a smooth curve, whereas in the left chart it has a sharp corner. The notion of a differentiable manifold refines that of a manifold by requiring the functions that transform between charts to be differentiable. Main article: History of manifolds and varieties.

This section needs expansion. You can help by adding to it. June Further information: tangent bundle. Further information: cotangent bundle.

### A Workbook for Students and Teachers

Further information: tensor bundle. Further information: frame bundle. Further information: jet bundle.

Further information: differential form. Further information: symplectic manifold. Further information: Lie group. Affine connection Atlas topology Christoffel symbols Introduction to the mathematics of general relativity List of formulas in Riemannian geometry Riemannian geometry Space mathematics. Note that a symplectic structure requires an additional integrability condition, beyond this isomorphism of groups: it is not just a G-structure.

Riemann Ricci , G. Ricci and T. Levi-Civita , T. Levi-Civita Weyl Whitney For an equivalent, ad hoc definition, see Sternberg Chapter II. Kobayashi Donaldson Milnor These are the first examples of exotic spheres. Sela However, 3-manifolds are only classified in the sense that there is an impractical algorithm for generating a non-redundant list of all compact 3-manifolds.