Motivation from surface theory

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.

Motivation from tensor calculus

Historically, people used the covariant derivative (or Levi-Civita connection given by the metric) to describe the variation rate of a vector along the direction of another vector. Here on the punctured 2-dimensional Euclidean space, the blue vector field X sends the one-form dr to 0.07 everywhere. The red vector field Y sends the one-form rdθ to 0.5r everywhere. Endorsed by the metric ds² = dr² + r²dθ², the Levi-Civita connection ∇_YX is 0 everywhere, indicating X has no change along Y. In other words, X parallel transports along each concentric circle. ∇_XY = Y/r everywhere, which sends rdθ to 0.5 everywhere, implying Y has a "constant" changing rate on the radial direction.

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,^[5] who developed a detailed mathematical foundation for general relativity, and Élie Cartan,^[6] who made the link with the geometrical ideas coming from surface theory.

Approaches

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.^{[lower-alpha 3]} 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.

Elementary properties

• 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 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 ∇¹
  _XY − ∇²
  _XY may be written Γ_x(X_x, Y_x) where
  $${\displaystyle \Gamma _{x}:\mathrm {T} _{x}M\times \mathrm {T} _{x}M\to \mathrm {T} _{x}M}$$

  
  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 Rⁿ, then the tangent bundle of M is the trivial bundle M × Rⁿ. 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 Rⁿ; then d_XY is the vector field corresponding to the smooth function dV(X) = ∂_XY from M to Rⁿ. Any other affine connection ∇ on M may therefore be written ∇ = d + Γ, 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 × Rⁿ. 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.

Explanations and historical intuition

The points just raised are easiest to explain in reverse, starting from the motivation provided by surface theory. In this situation, although the planes being rolled over the surface are tangent planes in a naive sense, the notion of a tangent space is really an infinitesimal notion,^{[lower-alpha 5]} whereas the planes, as affine subspaces of R³, are infinite in extent. However these affine planes all have a marked point, the point of contact with the surface, and they are tangent to the surface at this point. The confusion therefore arises because an affine space with a marked point can be identified with its tangent space at that point. However, the parallel transport defined by rolling does not fix this origin: it is affine rather than linear; the linear parallel transport can be recovered by applying a translation.

Abstracting this idea, an affine manifold should therefore be an n-manifold M with an affine space A_x, of dimension n, attached to each x ∈ M at a marked point a_x ∈ A_x, together with a method for transporting elements of these affine spaces along any curve C in M. This method is required to satisfy several properties:

1. for any two points x, y on C, parallel transport is an affine transformation from A_x to A_y;
2. parallel transport is defined infinitesimally in the sense that it is differentiable at any point on C and depends only on the tangent vector to C at that point;
3. the derivative of the parallel transport at x determines a linear isomorphism from T_xM to T_axA_x.

These last two points are quite hard to make precise,^[9] so affine connections are more often defined infinitesimally. To motivate this, it suffices to consider how affine frames of reference transform infinitesimally with respect to parallel transport. (This is the origin of Cartan's method of moving frames.) An affine frame at a point consists of a list (p, e₁,… e_n), where p ∈ A_x^{[lower-alpha 6]} and the e_i form a basis of T_p(A_x). The affine connection is then given symbolically by a first order differential system

${\displaystyle (*){\begin{cases}\mathrm {d} {p}&=\theta ^{1}\mathbf {e} _{1}+\cdots +\theta ^{n}\mathbf {e} _{n}\\\mathrm {d} \mathbf {e} _{i}&=\omega _{i}^{1}\mathbf {e} _{1}+\cdots +\omega _{i}^{n}\mathbf {e} _{n}\end{cases}}\quad i=1,2,\ldots ,n}$

defined by a collection of one-forms (θ ^j, ω ^j
_i). Geometrically, an affine frame undergoes a displacement travelling along a curve γ from γ(t) to γ(t + δt) given (approximately, or infinitesimally) by

${\displaystyle {\begin{aligned}p(\gamma (t+\delta t))-p(\gamma (t))&=\left(\theta ^{1}\left(\gamma '(t)\right)\mathbf {e} _{1}+\cdots +\theta ^{n}\left(\gamma '(t)\right)\mathbf {e} _{n}\right)\mathrm {\delta } t\\\mathbf {e} _{i}(\gamma (t+\delta t))-\mathbf {e} _{i}(\gamma (t))&=\left(\omega _{i}^{1}\left(\gamma '(t)\right)\mathbf {e} _{1}+\cdots +\omega _{i}^{n}\left(\gamma '(t)\right)\mathbf {e} _{n}\right)\delta t\,.\end{aligned}}}$

Furthermore, the affine spaces A_x are required to be tangent to M in the informal sense that the displacement of a_x along γ can be identified (approximately or infinitesimally) with the tangent vector γ′(t) to γ at x = γ(t) (which is the infinitesimal displacement of x). Since

${\displaystyle a_{x}(\gamma (t+\delta t))-a_{x}(\gamma (t))=\theta \left(\gamma '(t)\right)\delta t\,,}$

where θ is defined by θ(X) = θ¹(X)e₁ + … + θⁿ(X)e_n, this identification is given by θ, so the requirement is that θ should be a linear isomorphism at each point.

The tangential affine space A_x is thus identified intuitively with an infinitesimal affine neighborhood of x.

The modern point of view makes all this intuition more precise using principal bundles (the essential idea is to replace a frame or a variable frame by the space of all frames and functions on this space). It also draws on the inspiration of Felix Klein's Erlangen programme,^[10] in which a geometry is defined to be a homogeneous space. Affine space is a geometry in this sense, and is equipped with a flat Cartan connection. Thus a general affine manifold is viewed as curved deformation of the flat model geometry of affine space.

Affine space as the flat model geometry

General affine geometries: formal definitions

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 (θ ⁱ, ω ^k
_j) 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 A = Aff(n)/GL(n) is an affine space of the same dimension.

Curvature and torsion

Curvature and torsion are the main invariants of an affine connection. As there are many equivalent ways to define the notion of an affine connection, so there are many different ways to define curvature and torsion.

From the Cartan connection point of view, the curvature is the failure of the affine connection η to satisfy the Maurer–Cartan equation

${\displaystyle \mathrm {d} \eta +{\tfrac {1}{2}}[\eta \wedge \eta ]=0,}$

where the second term on the left hand side is the wedge product using the Lie bracket in aff(n) to contract the values. By expanding η into the pair (θ, ω) and using the structure of the Lie algebra aff(n), this left hand side can be expanded into the two formulae

${\displaystyle \mathrm {d} \theta +\omega \wedge \theta \quad {\text{and}}\quad \mathrm {d} \omega +\omega \wedge \omega \,,}$

where the wedge products are evaluated using matrix multiplication. The first expression is called the torsion of the connection, and the second is also called the curvature.

These expressions are differential 2-forms on the total space of a frame bundle. However, they are horizontal and equivariant, and hence define tensorial objects. These can be defined directly from the induced covariant derivative ∇ on TM as follows.

The torsion is given by the formula

${\displaystyle T^{\nabla }(X,Y)=\nabla _{X}Y-\nabla _{Y}X-[X,Y].}$

If the torsion vanishes, the connection is said to be torsion-free or symmetric.

The curvature is given by the formula

${\displaystyle R_{X,Y}^{\nabla }Z=\nabla _{X}\nabla _{Y}Z-\nabla _{Y}\nabla _{X}Z-\nabla _{[X,Y]}Z.}$

Note that [X, Y] is the Lie bracket of vector fields

${\displaystyle [X,Y]=\left(X^{j}\partial _{j}Y^{i}-Y^{j}\partial _{j}X^{i}\right)\partial _{i}}$

in Einstein notation. This is independent of coordinate system choice and

${\displaystyle \partial _{i}=\left({\frac {\partial }{\partial \xi ^{i}}}\right)_{p}\,,}$

the tangent vector at point p of the ith coordinate curve. The ∂_i are a natural basis for the tangent space at point p, and the X ⁱ the corresponding coordinates for the vector field X = X ⁱ ∂_i.

When both curvature and torsion vanish, the connection defines a pre-Lie algebra structure on the space of global sections of the tangent bundle.

The Levi-Civita connection

If (M, g) is a Riemannian manifold then there is a unique affine connection ∇ on M with the following two properties:

• the connection is torsion-free, i.e., T^∇ is zero, so that ∇_XY − ∇_YX = [X, Y];
• parallel transport is an isometry, i.e., the inner products (defined using g) between tangent vectors are preserved.

This connection is called the Levi-Civita connection.

The term "symmetric" is often used instead of torsion-free for the first property. The second condition means that the connection is a metric connection in the sense that the Riemannian metric g is parallel: ∇g = 0. For a torsion-free connection, the condition is equivalent to the identity X g(Y, Z) = g(∇_XY, Z) + g(Y, ∇_X Z), "compatibility with the metric".^[11] In local coordinates the components of the form are called Christoffel symbols: because of the uniqueness of the Levi-Civita connection, there is a formula for these components in terms of the components of g.

Geodesics

Since straight lines are a concept in affine geometry, affine connections define a generalized notion of (parametrized) straight lines on any affine manifold, called affine geodesics. Abstractly, a parametric curve γ : I → M is a straight line if its tangent vector remains parallel and equipollent with itself when it is transported along γ. From the linear point of view, an affine connection M distinguishes the affine geodesics in the following way: a smooth curve γ : I → M is an affine geodesic if ${\displaystyle {\dot {\gamma }}}$ is parallel transported along γ, that is

${\displaystyle \tau _{t}^{s}{\dot {\gamma }}(s)={\dot {\gamma }}(t)}$

where τ^s
_t : T_γsM → T_γtM is the parallel transport map defining the connection.

In terms of the infinitesimal connection ∇, the derivative of this equation implies

${\displaystyle \nabla _{{\dot {\gamma }}(t)}{\dot {\gamma }}(t)=0}$

for all t ∈ I.

Conversely, any solution of this differential equation yields a curve whose tangent vector is parallel transported along the curve. For every x ∈ M and every X ∈ T_xM, there exists a unique affine geodesic γ : I → M with γ(0) = x and γ̇(0) = X and where I is the maximal open interval in R, containing 0, on which the geodesic is defined. This follows from the Picard–Lindelöf theorem, and allows for the definition of an exponential map associated to the affine connection.

In particular, when M is a (pseudo-)Riemannian manifold and ∇ is the Levi-Civita connection, then the affine geodesics are the usual geodesics of Riemannian geometry and are the locally distance minimizing curves.

The geodesics defined here are sometimes called affinely parametrized, since a given straight line in M determines a parametric curve γ through the line up to a choice of affine reparametrization γ(t) → γ(at + b), where a and b are constants. The tangent vector to an affine geodesic is parallel and equipollent along itself. An unparametrized geodesic, or one which is merely parallel along itself without necessarily being equipollent, need only satisfy

${\displaystyle \nabla _{\dot {\gamma }}{\dot {\gamma }}=k{\dot {\gamma }}}$

for some function k defined along γ. Unparametrized geodesics are often studied from the point of view of projective connections.

Development

An affine connection defines a notion of development of curves. Intuitively, development captures the notion that if x_t 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 C_t in this affine space: the development of x_t.

In formal terms, let τ⁰
_t : T_xtM → T_x0M be the linear parallel transport map associated to the affine connection. Then the development C_t is the curve in T_x0M starts off at 0 and is parallel to the tangent of x_t for all time t:

${\displaystyle {\dot {C}}_{t}=\tau _{t}^{0}{\dot {x}}_{t}\,,\quad C_{0}=0.}$

In particular, x_t is a geodesic if and only if its development is an affinely parametrized straight line in T_x0M.^[12]

Example: the unit sphere in Euclidean space

Let ⟨ , ⟩ be the usual scalar product on R³, and let S² be the unit sphere. The tangent space to S² at a point x is naturally identified with the vector subspace of R³ consisting of all vectors orthogonal to x. It follows that a vector field Y on S² can be seen as a map Y : S² → R³ which satisfies

${\displaystyle \langle Y_{x},x\rangle =0\,,\quad \forall x\in \mathbf {S} ^{2}.}$

Denote as dY the differential (Jacobian matrix) of such a map. Then we have:

Lemma. The formula

${\displaystyle (\nabla _{Z}Y)_{x}=\mathrm {d} Y_{x}(Z_{x})+\langle Z_{x},Y_{x}\rangle x}$

defines an affine connection on S² with vanishing torsion.

Proof. It is straightforward to prove that ∇ satisfies the Leibniz identity and is C^∞(S²) linear in the first variable. So all that needs to be proved here is that the map above does indeed define a tangent vector field. That is, we need to prove that for all x in S²

${\displaystyle {\bigl \langle }(\nabla _{Z}Y)_{x},x{\bigr \rangle }=0\,.\qquad {\text{(Eq.1)}}}$

Consider the map

${\displaystyle {\begin{aligned}f:\mathbf {S} ^{2}&\to \mathbf {R} \\x&\mapsto \langle Y_{x},x\rangle \,.\end{aligned}}}$

The map f is constant, hence its differential vanishes. In particular

${\displaystyle \mathrm {d} f_{x}(Z_{x})={\bigl \langle }(\mathrm {d} Y)_{x}(Z_{x}),x(\gamma '(t)){\bigr \rangle }+\langle Y_{x},Z_{x}\rangle =0\,.}$

Equation 1 above follows. Q.E.D.

Primary historical references

Secondary references