Манифольдтардың жанама векторлық өрістерін дифференциалдау құрылысы

Construct allowing differentiation of tangent vector fields of manifolds

In differential geometry, an affine connection is a geometric object on a smooth manifold which connects nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space. Connections are among the simplest methods of defining differentiation of the sections of vector bundles. The notion of an affine connection has its roots in 19th century geometry and tensor calculus, but was not fully developed until the early 1920s, by Élie Cartan (as part of his general theory of connections) and Hermann Weyl (who used the notion as a part of his foundations for general relativity). The terminology is due to Cartan and has its origins in the identification of tangent spaces in Euclidean space 'R'^(n) by translation: the idea is that a choice of affine connection makes a manifold look infinitesimally like Euclidean space not just smoothly, but as an affine space. On any manifold of positive dimension there are infinitely many affine connections. If the manifold is further endowed with a metric tensor then there is a natural choice of affine connection, called the Levi Civita connection. The choice of an affine connection is equivalent to prescribing a way of differentiating vector fields which satisfies several reasonable properties (linearity and the Leibniz rule). This yields a possible definition of an affine connection as a covariant derivative or (linear) connection on the tangent bundle. A choice of affine connection is also equivalent to a notion of parallel transport, which is a method for transporting tangent vectors along curves. This also defines a parallel transport on the frame bundle. Infinitesimal parallel transport in the frame bundle yields another description of an affine connection, either as a Cartan connection for the affine group or as a principal connection on the frame bundle. The main invariants of an affine connection are its torsion and its curvature. The torsion measures how closely the Lie bracket of vector fields can be recovered from the affine connection. Affine connections may also be used to define (affine) geodesics on a manifold, generalizing the straight lines of Euclidean space, although the geometry of those straight lines can be very different from usual Euclidean geometry; the main differences are encapsulated in the curvature of the connection.

A smooth manifold is a mathematical object which looks locally like a smooth deformation of Euclidean space 'R'^(n): for example a smooth curve or surface looks locally like a smooth deformation of a line or a plane. Smooth functions and vector fields can be defined on manifolds, just as they can on Euclidean space, and scalar functions on manifolds can be differentiated in a natural way. However, differentiation of vector fields is less straightforward: this is a simple matter in Euclidean space, because the tangent space of based vectors at a point p can be identified naturally (by translation) with the tangent space at a nearby point q. On a general manifold, there is no such natural identification between nearby tangent spaces, and so tangent vectors at nearby points cannot be compared in a well defined way. The notion of an affine connection was introduced to remedy this problem by connecting nearby tangent spaces. The origins of this idea can be traced back to two main sources: surface theory and tensor calculus.

Consider a smooth surface S in a 3 dimensional Euclidean space. Near any point, S can be approximated by its tangent plane at that point, which is an affine subspace of Euclidean space. Differential geometers in the 19th century were interested in the notion of development in which one surface was rolled along another, without slipping or twisting. In particular, the tangent plane to a point of S can be rolled on S: this should be easy to imagine when S is a surface like the 2 sphere, which is the smooth boundary of a convex region. As the tangent plane is rolled on S, the point of contact traces out a curve on S. Conversely, given a curve on S, the tangent plane can be rolled along that curve. This provides a way to identify the tangent planes at different points along the curve: in particular, a tangent vector in the tangent space at one point on the curve is identified with a unique tangent vector at any other point on the curve. These identifications are always given by affine transformations from one tangent plane to another. This notion of parallel transport of tangent vectors, by affine transformations, along a curve has a characteristic feature: the point of contact of the tangent plane with the surface always moves with the curve under parallel translation (i. e., as the tangent plane is rolled along the surface, the point of contact moves). This generic condition is characteristic of Cartan connections. In more modern approaches, the point of contact is viewed as the origin in the tangent plane (which is then a vector space), and the movement of the origin is corrected by a translation, so that parallel transport is linear, rather than affine. In the point of view of Cartan connections, however, the affine subspaces of Euclidean space are model surfaces — they are the simplest surfaces in Euclidean 3 space, and are homogeneous under the affine group of the plane — and every smooth surface has a unique model surface tangent to it at each point. These model surfaces are Klein geometries in the sense of Felix Klein's Erlangen programme. More generally, an n dimensional affine space is a Klein geometry for the affine group Aff(n), the stabilizer of a point being the general linear group GL(n). An affine n manifold is then a manifold which looks infinitesimally like n dimensional affine space.

The second motivation for affine connections comes from the notion of a covariant derivative of vector fields. Before the advent of coordinate independent methods, it was necessary to work with vector fields by embedding their respective Euclidean vectors into an atlas. These components can be differentiated, but the derivatives do not transform in a manageable way under changes of coordinates. Correction terms were introduced by Elwin Bruno Christoffel (following ideas of Bernhard Riemann) in the 1870s so that the (corrected) derivative of one vector field along another transformed covariantly under coordinate transformations — these correction terms subsequently came to be known as Christoffel symbols. This idea was developed into the theory of absolute differential calculus (now known as tensor calculus) by Gregorio Ricci Curbastro and his student Tullio Levi Civita between 1880 and the turn of the 20th century. Tensor calculus really came to life, however, with the advent of Albert Einstein's theory of general relativity in 1915. A few years after this, Levi Civita formalized the unique connection associated to a Riemannian metric, now known as the Levi Civita connection. More general affine connections were then studied around 1920, by Hermann Weyl, who developed a detailed mathematical foundation for general relativity, and Élie Cartan, who made the link with the geometrical ideas coming from surface theory.

The complex history has led to the development of widely varying approaches to and generalizations of the affine connection concept. The most popular approach is probably the definition motivated by covariant derivatives. On the one hand, the ideas of Weyl were taken up by physicists in the form of gauge theory and gauge covariant derivatives. On the other hand, the notion of covariant differentiation was abstracted by Jean Louis Koszul, who defined (linear or Koszul) connections on vector bundles. In this language, an affine connection is simply a covariant derivative or (linear) connection on the tangent bundle. However, this approach does not explain the geometry behind affine connections nor how they acquired their name. The term really has its origins in the identification of tangent spaces in Euclidean space by translation: this property means that Euclidean n space is an affine space. (Alternatively, Euclidean space is a principal homogeneous space or torsor under the group of translations, which is a subgroup of the affine group.) As mentioned in the introduction, there are several ways to make this precise: one uses the fact that an affine connection defines a notion of parallel transport of vector fields along a curve. This also defines a parallel transport on the frame bundle. Infinitesimal parallel transport in the frame bundle yields another description of an affine connection, either as a Cartan connection for the affine group Aff(n) or as a principal GL(n) connection on the frame bundle.

It follows from property 1 above that the value of ∇XY at a point x ∈ M depends only on the value of X at x and not on the value of X on 1=M − {x}. It also follows from property 2 above that the value of ∇XY at a point x ∈ M depends only on the value of Y on a neighbourhood of x. If ∇^(1), ∇^(2) are affine connections then the value at x of may be written Γx(Xx, Yx) where is bilinear and depends smoothly on x (i. e., it defines a smooth bundle homomorphism). Conversely if ∇ is an affine connection and Γ is such a smooth bilinear bundle homomorphism (called a connection form on M) then ∇ + Γ is an affine connection. If M is an open subset of 'R'^(n), then the tangent bundle of M is the trivial bundle M × 'R'^(n). In this situation there is a canonical affine connection d on M: any vector field Y is given by a smooth function V from M to 'R'^(n); then dXY is the vector field corresponding to the smooth function from M to 'R'^(n). Any other affine connection ∇ on M may therefore be written , where Γ is a connection form on M.
More generally, a local trivialization of the tangent bundle is a bundle isomorphism between the restriction of TM to an open subset U of M, and U × 'R'^(n). The restriction of an affine connection ∇ to U may then be written in the form d + Γ where Γ is a connection form on U.

Informally, an affine space is a vector space without a fixed choice of origin. It describes the geometry of points and free vectors in space. As a consequence of the lack of origin, points in affine space cannot be added together as this requires a choice of origin with which to form the parallelogram law for vector addition. However, a vector v may be added to a point p by placing the initial point of the vector at p and then transporting p to the terminal point. The operation thus described p → p + v is the translation of p along v. In technical terms, affine n space is a set 'A'^(n) equipped with a free transitive action of the vector group 'R'^(n) on it through this operation of translation of points: 'A'^(n) is thus a principal homogeneous space for the vector group 'R'^(n). The general linear group GL(n) is the group of transformations of 'R'^(n) which preserve the linear structure of 'R'^(n) in the sense that By analogy, the affine group Aff(n) is the group of transformations of 'A'^(n) preserving the affine structure. Thus φ ∈ Aff(n) must preserve translations in the sense that

where T is a general linear transformation. The map sending φ ∈ Aff(n) to T ∈ GL(n) is a group homomorphism. Its kernel is the group of translations 'R'^(n). The stabilizer of any point p in A can thus be identified with GL(n) using this projection: this realises the affine group as a semidirect product of GL(n) and 'R'^(n), and affine space as the homogeneous space Aff(n)/GL(n).

An affine space, as with essentially any smooth Klein geometry, is a manifold equipped with a flat Cartan connection. More general affine manifolds or affine geometries are obtained easily by dropping the flatness condition expressed by the Maurer Cartan equations. There are several ways to approach the definition and two will be given. Both definitions are facilitated by the realisation that 1 forms in the flat model fit together to give a 1 form with values in the Lie algebra 'aff'(n) of the affine group Aff(n). In these definitions, M is a smooth n manifold and is an affine space of the same dimension.

An affine connection on M is a principal Aff(n) bundle Q over M, together with a principal GL(n) subbundle P of Q and a principal Aff(n) connection α (a 1 form on Q with values in 'aff'(n)) which satisfies the following (generic) Cartan condition. The 'R'^(n) component of pullback of α to P is a horizontal equivariant 1 form and so defines a bundle homomorphism from TM to P ×GL(n) 'R'^(n): this is required to be an isomorphism.

Since Aff(n) acts on A, there is, associated to the principal bundle Q, a bundle , which is a fiber bundle over M whose fiber at x in M is an affine space Ax. A section a of A (defining a marked point ax in Ax for each x ∈ M) determines a principal GL(n) subbundle P of Q (as the bundle of stabilizers of these marked points) and vice versa. The principal connection α defines an Ehresmann connection on this bundle, hence a notion of parallel transport. The Cartan condition ensures that the distinguished section a always moves under parallel transport.

An affine connection defines a notion of development of curves. Intuitively, development captures the notion that if xt is a curve in M, then the affine tangent space at x0 may be rolled along the curve. As it does so, the marked point of contact between the tangent space and the manifold traces out a curve Ct in this affine space: the development of xt. In formal terms, let be the linear parallel transport map associated to the affine connection. Then the development Ct is the curve in Tx0M starts off at 0 and is parallel to the tangent of xt for all time t:

In particular, xt is a geodesic if and only if its development is an affinely parametrized straight line in Tx0M.

If M is a surface in 'R'^(3), it is easy to see that M has a natural affine connection. From the linear connection point of view, the covariant derivative of a vector field is defined by differentiating the vector field, viewed as a map from M to 'R'^(3), and then projecting the result orthogonally back onto the tangent spaces of M. It is easy to see that this affine connection is torsion free. Furthermore, it is a metric connection with respect to the Riemannian metric on M induced by the inner product on 'R'^(3), hence it is the Levi Civita connection of this metric.

Cartan's treatment of affine connections as motivated by the study of relativity theory. Includes a detailed discussion of the physics of reference frames, and how the connection reflects the physical notion of transport along a worldline. A more mathematically motivated account of affine connections. Affine connections from the point of view of Riemannian geometry. Robert Hermann's appendices discuss the motivation from surface theory, as well as the notion of affine connections in the modern sense of Koszul. He develops the basic properties of the differential operator ∇, and relates them to the classical affine connections in the sense of Cartan.