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 Rⁿ 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 Rⁿ. 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.

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:

F(x, my) = mF(x,y) for all x, y in TM,
F is infinitely differentiable in TM − {0},
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 $det(M) = ( - 1) n det(M)$ 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 $g (J X, J Y) = g (X, Y)$ . An almost hermitian structure defines naturally a 2- differential form $ω J, g (X, Y): = g (J X, Y)$ The following two conditions are equivalent:

1) $N J = 0 and d ω = 0;$

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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