Differentiable manifold

The emergence of differential geometry as a distinct discipline is generally credited to Carl Friedrich Gauss and Bernhard Riemann.

[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: The works of physicists such as James Clerk Maxwell,[2] and mathematicians Gregorio Ricci-Curbastro and Tullio Levi-Civita[3] 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.

[4] The widely accepted general definition of a manifold in terms of an atlas is due to Hassler Whitney.

For this reason, one refers separately to the "real" and "complex" dimension of a topological space with a holomorphic atlas.

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.

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.

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.

Given a real valued function f on an n dimensional differentiable manifold M, the directional derivative of f at a point p in M is defined as follows.

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.

Let {Uα} be an open covering of M. Then a partition of unity subordinate to the cover {Uα} is a collection of real-valued Ck functions φi on M satisfying the following conditions: (Note that this last condition is actually a finite sum at each point because of the local finiteness of the supports of the φi.)

This allows for certain constructions from the topology of Ck functions on Rn to be carried over to the category of differentiable manifolds.

A moving tangent frame is an ordered list of vector fields that give a basis at every point of their domain.

These and other examples of the general idea of jet bundles play a significant role in the study of differential operators on manifolds.

Differentiable functions between two manifolds are needed in order to formulate suitable notions of submanifolds, and other related concepts.

which is not smooth relative to the given atlas; for instance, one can modify the identity map using a localized non-smooth bump.

A remarkable result, proved in 2002 by methods of partial differential equations, is the geometrization 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 as Mostow rigidity and Sela's algorithm for the isomorphism problem for hyperbolic groups.

These objects often arise naturally in describing (continuous) symmetries, and they form an important source of examples of smooth manifolds.

The notion of a pseudogroup[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.

Note that Γ need not be a group, however, since the functions are not globally defined on S. For example, the collection of all local Ck diffeomorphisms on Rn form a pseudogroup.

Thus, pseudogroups provide a single framework in which to describe many structures on manifolds of importance to differential geometry and topology.

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.

This means that [12] for each point p ∈ M, there is a neighborhood U of p, and a pair of functions (f, f#), where There are a number of important motivations for studying differentiable manifolds within this abstract framework.

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).

Rather, it emerges as a sheaf of functions as a consequence of the construction (via the quotients of local rings by their maximal ideals).

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.

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 the Banach–Stone theorem, and is more formally known as the spectrum of a C*-algebra.