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Differential geometry and topology

2007 Schools Wikipedia Selection. Related subjects: Mathematics

   In mathematics, differential topology is the field dealing with
   differentiable functions on differentiable manifolds. It arises
   naturally from the study of the theory of differential equations.
   Differential geometry is the study of geometry using differential
   calculus (cf. integral geometry). These fields are adjacent, and have
   many applications in physics, notably in the theory of relativity.
   Together they make up the geometric theory of differentiable manifolds
   - which can also be studied directly from the point of view of
   dynamical systems.

Intrinsic versus extrinsic

   Initially and up to the middle of the nineteenth century, differential
   geometry was studied from the extrinsic point of view: curves and
   surfaces were considered as lying in a Euclidean space of higher
   dimension (for example a surface in an ambient space of three
   dimensions). The simplest results are those in the differential
   geometry of curves. Starting with the work of Riemann, the intrinsic
   point of view was developed, in which one cannot speak of moving
   'outside' the geometric object because it is considered as given in a
   free-standing way.

   The intrinsic point of view is more flexible. For example, it is useful
   in relativity where space-time cannot naturally be taken as extrinsic
   (what would be 'outside' it?). With the intrinsic point of view it is
   harder to define the central concept of curvature and other structures
   such as connections, so there is a price to pay.

   These two points of view can be reconciled, i.e. the extrinsic geometry
   can be considered as a structure additional to the intrinsic one (see
   the Nash embedding theorem).

Technical requirements

   The apparatus of differential geometry is that of calculus on
   manifolds: this includes the study of manifolds, tangent bundles,
   cotangent bundles, differential forms, exterior derivatives, integrals
   of p-forms over p-dimensional submanifolds and Stokes' theorem, wedge
   products, and Lie derivatives. These all relate to multivariable
   calculus; but for the geometric applications must be developed in a way
   that makes good sense without a preferred coordinate system. The
   distinctive concepts of differential geometry can be said to be those
   that embody the geometric nature of the second derivative: the many
   aspects of curvature.

   A real differentiable manifold is a topological space with a collection
   of diffeomorphisms from open sets of the space to open subsets in R^n
   such that the open sets cover the space, and if f, g are
   diffeomorphisms then the composite mapping f o g ^-1 from an open
   subset of the open unit ball to the open unit ball is infinitely
   differentiable. We say a function from the manifold to R is infinitely
   differentiable if its composition with every diffeomorphism results in
   an infinitely differentiable function from the open unit ball to R. Of
   course manifolds need not be real, for example we can have complex
   manifolds.

   At every point of the manifold, there is the tangent space at that
   point, which consists of every possible velocity (direction and
   magnitude) with which it is possible to travel away from this point.
   For an n-dimensional manifold, the tangent space at any point is an
   n-dimensional vector space, or in other words a copy of R^n. The
   tangent space has many definitions. One definition of the tangent space
   is as the dual space to the linear space of all functions which are
   zero at that point, divided by the space of functions which are zero
   and have a first derivative of zero at that point. Having a zero
   derivative can be defined by "composition by every differentiable
   function to the reals has a zero derivative", so it is defined just by
   differentiability.

   A vector field is a function from a manifold to the disjoint union of
   its tangent spaces (this union is itself a manifold known as the
   tangent bundle), such that at each point, the value is an element of
   the tangent space at that point. Such a mapping is called a section of
   a bundle. A vector field is differentiable if for every differentiable
   function, applying the vector field to the function at each point
   yields a differentiable function. Vector fields can be thought of as
   time-independent differential equations. A differentiable function from
   the reals to the manifold is a curve on the manifold. This defines a
   function from the reals to the tangent spaces: the velocity of the
   curve at each point it passes through. A curve will be said to be a
   solution of the vector field if, at every point, the velocity of the
   curve is equal to the vector field at that point.

   An alternating k-dimensional linear form is an element of the
   antisymmetric k'th tensor power of the dual V^* of some vector space V.
   A differential k-form on a manifold is a choice, at each point of the
   manifold, of such an alternating k-form -- where V is the tangent space
   at that point. This will be called differentiable if whenever it
   operates on k differentiable vector fields, the result is a
   differentiable function from the manifold to the reals. A space form is
   a linear form with the dimensionality of the manifold.

Differential topology

   Differential topology per se considers the properties and structures
   that require only a smooth structure on a manifold to define (such as
   those in the previous section). Smooth manifolds are 'softer' than
   manifolds with extra geometric structures, which can act as
   obstructions to certain types of equivalences and deformations that
   exist in differential topology. For instance, volume and Riemannian
   curvature are invariants that can distinguish different geometric
   structures on the same smooth manifold—that is, one can smoothly
   "flatten out" certain manifolds, but it might require distorting the
   space and affecting the curvature or volume.

   Conversely, smooth manifolds are more rigid than the topological
   manifolds. Certain topological manifolds have no smooth structures at
   all (see Donaldson's theorem) and others have more than one
   inequivalent smooth structure (such as exotic spheres). Some
   constructions of smooth manifold theory, such as the existence of
   tangent bundles, can be done in the topological setting with much more
   work, and others cannot.

   See also transversality

Branches of differential geometry

Contact geometry

   Contact geometry is an analog of symplectic geometry which works for
   certain manifolds of odd dimension. Roughly, the contact structure on
   (2n+1)-dimensional manifold is a choice of a hyperplane field that is
   nowhere integrable. This is equivalent to the hyperplane field being
   defined by a 1-form α such that \alpha\wedge (d\alpha)^n does not
   vanish anywhere.

Finsler geometry

   Finsler geometry has the Finsler manifold as the main object of study —
   this is a differential manifold with a Finsler metric, i.e. a Banach
   norm defined on each tangent space. A Finsler metric is much more
   general structure than a Riemannian metric. Then a Finsler structure on
   a manifold M is a function F : TM → [0,n) such that:
    1. F(x, my) = mF(x,y) for all x, y in TM,
    2. F is infinitely differentiable in TM − {0},
    3. The vertical Hessian of FF/2 is positive definite.

Riemannian geometry

   Riemannian geometry has Riemannian manifolds as the main object of
   study — smooth manifolds with additional structure which makes them
   look infinitesimally like Euclidean space. These allow one to
   generalise the notion from Euclidean geometry and analysis such as
   gradient of a function, divergence, length of curves and so on; without
   assumptions that the space is globally so symmetric. The Riemannian
   curvature tensor is an important pointwise invariant associated to a
   Riemannian manifold that measures how close it is to being flat.

Symplectic geometry

   Symplectic geometry is the study of symplectic manifolds. A symplectic
   manifold is a differentiable manifold equipped with a symplectic form
   ω(that is, a non-degenerate, bilinear, skew-symmetric and closed 2-
   form). Since the symplectic form must be skew-symmetric, its matrix
   representation must be skew-symmetric, i.e.

          M^{\top} = -M .

   It follows that det(M) = ( − 1)^ndet(M) if M is an n \times n matrix.
   Thus, for odd n we see that det(M) = 0, and so non-degenerate
   skew-symmetric two forms can only exist on even dimensional spaces.
   Unlike in Riemannian geometry, all symplectic manifolds are locally
   isomorphic: this is called Darboux's theorem and follows from the
   assumption that ω is closed, so the only invariants of a symplectic
   manifold are global in nature. A diffeomorphism between two symplectic
   spaces which preserves the symplectic structure (i.e. the symplectic
   form) is called a symplectomorphism.

   In dimension 2, a symplectic manifold is just a manifold endowed with
   an area form. The first result in symplectic topology is probably the
   Poincare-Birkhoff theorem, conjectured by Poincare and proved by
   Birkhoff in 1912. This claims that if an area preserving map of a ring
   twists each boundary component in opposite directions, then the map has
   at least two fixed points.

   It is easy to show that the area preserving condition (or the twisting
   condition) cannot be removed.

   Note that if one tries to extend such a theorem to higher dimensions,
   one would probably guess that a volume preserving map of a certain type
   must have fixed points. This is false in dimensions greater than 3.

Complex and Kähler geometry

   Complex differential geometry is the study of complex manifolds. An
   almost complex manifold is a real manifold M, endowed with a tensor of
   type (1,1), i.e. a vector bundle endomorphism (called almost complex
   structure)

   J:TM\rightarrow TM , such that J^2 = − 1.

   By definition, an almost complex manifold is even dimensional.

   An almost complex manifold is called complex if N[J] = 0, where N[J] is
   a tensor of type (2,1) related to J, called "torsion" or "Nijenhuis
   tensor". An almost complex manifold is complex if and only if admits an
   holomorphic atlas. An almost hermitian structure is given by a couple
   (J,g) where J is an almost complex structure and g is a riemannian
   metric, satisfying the compatibility condition g(JX,JY) = g(X,Y). An
   almost hermitian structure defines naturally a 2- differential form
   ω[J,g](X,Y): = g(JX,Y) The following two conditions are equivalent:

   1) N[J] = 0 and dω = 0;

   2) \nabla J=0,

   where \nabla is the Levi-Civita connection of g. In this case, (J,g) is
   called a Kähler structure. In particular, a Kähler manifold is a
   manifold endowed with a Kähler structure. In particular, a Kähler
   manifold is a complex and a symplectic manifold. A large class of
   Kähler manifolds is given by all the smooth complex algebraic
   varieties.

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