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1 Introduction

How can we differentiate a section s in a vector bundle E over a manifold M? It takes values in different vector spaces E_p for each p\in M, but differentiating involves comparing values at different points p. This needs an extra structure on E which connects the different vector spaces E_p, p\in M, among each other; it is therefore called connection. It can be defined in two different ways. Either it is viewed as a linear map which assigns to each vector field X on M a so called covariant derivative, a differential operator \nabla_{\!X} on the space of sections on E. Or else it is given as a parallel displacement along curves in M: Given two points p,q\in M and a curve c connecting these points, a connection allows to move any element of E_p to E_q along c. Infinitesimally it is given by a distribution \mathcal{H} on E or on its frame bundle FE. Let us repeat: In ordinary analysis we differentiate functions with values in a constant vector space; differentiating functions with values in a variable vector space E_p (a vector bundle) needs an extra structure called connection. The greater generality leads to a new notion: curvature. In the covariant derivative model, curvature is the commutator [\nabla_i,\nabla_j] of covariant derivatives in two coordinate directions. For ordinary derivatives \nabla_i = \partial_i this quantity vanishes, and for covariant derivatives it is an algebraic quantity (a tensor) rather than a differential operator. In the parallel displacement model, curvature is just the non-integrability of the distribution, the tensor [\mathcal{H},\mathcal{H}]^{\mathcal{H}^\perp}. A case of particular importance is the tangent bundle, E = TM. A connection on TM yields a second tensor quantity beneath curvature, the so called torsion. This is explained best by passing to the affine frame bundle; therefore a connection on TM is sometimes called affine connection.

2 Covariant derivatives

A covariant derivative on a vector bundle E over a smooth manifold M is a directional derivative \nabla for sections of E. It can be viewed as a bilinear map \nabla : \Gamma TM \times \Gamma E \to \Gamma E, (X,s) \mapsto \nabla_Xs which is a tensor (linear over C^\infty(M)) in the first argument and a derivation in the second argument:

(1)\begin{matrix} \nabla_{(fX)}s &=& f\nabla_Xs\,\cr \nabla_{X}(fs) &=& (Xf)s + f\nabla_Xs \,, \end{matrix}

where f is a smooth function and X a vector field on M and s a section of E, and where Xf = \partial_Xf = df.X is the ordinary derivative of the function f in the direction of X. By these properties, \nabla is defined locally and even pointwise regarding the first argument: For any v\in T_pM we may define \nabla_xs := (\nabla_Xs)_p where X is any (local) vector field with X_p = x.

3 Curvature

A covariant derivative has all properties of the ordinary directional derivative for functions with exception of the commutativity: For functions f we have \partial_i\partial_jf = \partial_j\partial_if with \partial_if = \frac{\partial(f\circ \phi)}{\partial u_i} = \partial_{\phi_i}f, where \phi : \R^n\to M is a local diffeomorphism (local parametrization of M) and \phi_i := \partial_i\phi its i-th partial derivative. Instead, for covariant derivatives \nabla_i = \nabla_{\phi_i} of a section s on a vector bundle E, the quantity

(2)R_{ij}s := [\nabla_i,\nabla_j]s = \nabla_i\nabla_js - \nabla_j\nabla_is

is in general nonzero but just a tensor (rather than a differential operator): R_{ij}(fs) = fR_{ij}s. For arbitrary vector fields X,Y with X \circ \phi = \sum_i\xi^i\phi_i and Y \circ \phi = \sum_j\eta^j\phi_j we put

(3)R(X,Y)s = \sum_{ij} \xi^i\eta^j R_{ij}s = [\nabla_X,\nabla_Y]s - \nabla_{[X,Y]}s

where [X,Y] = \sum_i \left((X\eta^i) - (Y\xi^i)\right)\phi_i denotes the Lie bracket of vector fields.

4 Moving frames

A moving frame on E is an n-tuple of sections s_1,\dots,s_n \in \Gamma E|_{M_o} on some open subset M_o \subset M such that for any p\in M_o the vectors s_1(p),\dots,s_n(p) form a basis of E_p. Denoting by \nabla s_j the linear map X \mapsto \nabla_Xs_j, we have

(4)\nabla s_j = \sum_i\omega_{ij} s_i
for certain 1-forms \omega_{ij}\in \Omega^1(M_o) called (local) connection forms. Differentiating a second time we have \nabla_X\nabla_Ys_j = \sum_i \left\{X\omega_{ij}(Y)s_i + \omega_{ij}(Y)\sum_{k}\omega_{ki}(X)s_k\right\} and therefore (interchanging the roles of the indices k and i in the second term)
(5)R(.,.)s_j = \sum_i \left(d\omega_{ij} + \sum_k \omega_{ik}\wedge \omega_{kj}\right)s_i

where we have used d\omega(X,Y) = X\omega(Y) - Y\omega(X) - \omega([X,Y]) and (\omega\wedge\psi)(X,Y) = \omega(X)\psi(Y)-\omega(Y)\psi(X) for arbitrary 1-forms \omega,\psi \in \Omega^1(M_o). On the other hand we let

(6)R(.,.)s_j = \Omega_{ij}s_i

for some 2-forms \Omega_{ji} \in \Omega^2(M_o) called (local) curvature forms, and thus we obtain the following relation between connection and curvature forms:

(7)\Omega_{ij} = d\omega_{ij} + \sum_k \omega_{ik}\wedge \omega_{kj}

called Cartan Structure equations. Using matrix notation \Omega = (\Omega_{ij}), \omega = (\omega_{ij}), we may write the latter equation in the short form

(8)\Omega = d\omega + \omega\wedge\omega.

5 Parallelity and connection

A section s defined along an injective smooth curve c : I \to M is called parallel if \nabla_{c'}s = 0. This is a linear ODE for s which is solvable along c with arbitrary initial values. If s_1,\dots,s_n is a basis of parallel sections and s = \sum_i\sigma_is_i is an arbitrary section along c, then \nabla_{c'}s = \sum_i \sigma_i's_i. Thus a covariant derivative is determined by its notion of parallelity and vice versa. Parallelity is given by a distribution \mathcal H on E, i.e.\ a subbundle \mathcal{H} \subset TE, where \mathcal{H}_v for v\in E_p consists of the initial vectors s'(a) of parallel sections s with s(a) = v along curves c : [a,b] \to M starting at c(a) = p. Stated differently, \mathcal{H}_v = ds_p(T_pM) where s is a local section defined near p such that s(p)=v and \nabla_xs = 0 for all x\in T_pM. Vice versa, a section s along any curve c (a smooth map s : I \to E with s(t) \in E_{c(t)} for all t\in I) is parallel iff s'(t) \in \mathcal{H}_{s(t)} for all t\in I. Since this distribution "connects" the distinct fibres of E among each other, it is called a "connection".

For any piecewise smooth curve c : [a,b] \to M from p to q and any initial value v \in E_p we have a parallel section s along c with s(a) = v. The mapping \tau_c : s(a) \mapsto s(b) : E_p \to E_q is an invertible linear map called parallel transport along the curve c. In general, parallel transport depends on the curve c itself, not only on the end points p,q, but it is independent of the parametrization of c. This dependence is measured by the holonomy group \mathcal{H}ol_p at p which is the set of parallel transports \tau_c along all loops c at p, i.e.piecewise smooth curves c : [a,b] \to M with c(a) = c(b) = p. It is known by the Ambrose-Singer theorem [Kobayashi&Nomizu1963, Theorem 8.1] that the connected component of \mathcal{H}ol_p \subset GL(E_p) is a Lie subgroup and its Lie algebra is spanned by the linear maps \tau_c^{-1}R(x,y)\tau_c for all curves c : [a,b] \to M starting from p and all x,y \in T_qM where q = c(b).

6 Connection on the frame bundle

Since it is useful to work with frames instead of single sections, we may replace E with the linear frame bundle FE whose fibre FE_p over p\in M is the set of all frames (bases) of the vector space E_p. Then parallelity of a frame (s_1,\dots,s_n), i.e. parallelity of all sections s_i in this frame, is expressed by a distribution on FE which is also called \mathcal H. Together with the the "vertical space" \mathcal V_f, the tangent space T_f(FE_p) of the fibre through f, it yields a direct decomposition T_fFE = \mathcal{H}_f \oplus \mathcal{V}_f and hence we have projections \pi_{\mathcal H} and \pi_{\mathcal V} of TFE onto the two subbundles. FE is a principal fibre bundle for the group G = GL_n, i.e.the fibres are the orbits of a free action of G from the right given by (f,g) \mapsto fg where f = (f_1,\dots,f_n) is a frame and g = (g_{ij}) a matrix and where fg is the line with j-th component \sum_i f_ig_{ij}. Fixing f\in FE_p, the action \phi_f : G \to FE_p, g\mapsto fg is a diffeomorphism which is equivariant with respect to right translation, and the left invariant vector fields on G are turned by \phi_* into vector fields on FE tangent to the fibres, so called fundamental vector fields. In particular, the vertical space \mathcal{V}_f is canonically isomorphic to the Lie algebra \frak{g} of G, via the infinitesimal action (\phi_f)_* = d(\phi_f)_e. Using this identification, the projection \pi_{\mathcal V} is a linear form on TFE with values in \frak{g}; it will be called (global) connection form \omega. The form \omega in Section 4 will be better called \omega_{\textup{f}} since it depends on a moving frame \textup{f}= (s_1,\dots,s_n) : M_o \to FE. We have

(9)\omega_{\textup{f}} = \textup{f}^*\omega.

7 Curvature on the frame bundle

We get the same Cartan structure equations as in Section 4

(10)d\omega + \omega\wedge\omega = \Omega

where the (global) curvature form \Omega\in \Omega^2(FE) is given by

(11)\Omega(U,V) = -\omega([\pi_{\mathcal{H}}U,\pi_{\mathcal{H}}V])

for all vector fields U,V on FE. To prove (10) we work with two special kinds of vector fields on FE, the fundamental vertical fields A,B etc. which are of type A_f = (d\phi_f)_ea = \frac{d}{dt} (f\exp ta)|_{t=0} for some a\in\frak{g}, and the horizontal lifts X,Y etc. which are horizontal vector fields projecting to a constant tangent vector on M along every fibre; they are \pi-related to a vector field \bar X on M. Choosing U = A+X and similarly V = B+Y, we have \omega(U) = a and \omega(V) = b and hence U\omega(V) = V\omega(U) = 0 since a,b are constant elements of \frak{g}. What remains is

\displaystyle -d\omega(U,V) = \omega([U,V]) = \omega([A,B] + [X,Y]) = [a,b] + \omega([X,Y])= [\omega(U),\omega(V)] - \Omega(U,V),
using (11). Here we see the curvature form in a new role: it measures the non-integrability of the horizontal distribution: \mathcal{H} is integrable \iff [X,Y] is horizontal \iff \Omega(X,Y) = -\omega([X,Y]) = 0.

8 Connections on general principal bundles

More generally, let P be a G-principal bundle over M: A manifold P with a smooth submersion \pi : P \to M and a free action of a Lie group G on P from the right such that the orbits are precisely the fibres, the preimages \pi^{-1}(p), p\in M. A connection on P is a G-invariant distribution \mathcal{H} on P (also called the "horizontal distribution") which is complementary to the tangent spaces of the fibres forming the "vertical distribution" \mathcal{V}. As before, each vertical space can be identified with the Lie algebra \frak{g} of G, and thus the vertical projection \pi_\mathcal{V} can be viewed as a \frak{g}-valued 1-form \omega \in \Omega(P,\frak{g}), and the equations (10) and (11) hold accordingly. If \Omega = 0, then P splits geometrically as G\times M at least locally, and (10) becomes the Maurer-Cartan equation of the Lie group G. For any smooth action \rho : G\times E_o \to E_o of G on a smooth manifold E_o we consider the associated bundle E = (P\times E_o)/G where the action on P\times E_o is given by g(p,v) = (pg^{-1},\rho(g)v). This is a bundle over M with fibre E_o, and since the distribution \mathcal{H} is G-invariant, it can be transferred to E via P \subset P\times E_o \to E, defining a connection on E. In the case P = FE for a vector bundle E and E_o = \R^n with its linear GL_n-action we have E \cong (PE\times\R^n)/GL_n, using the map PE \times \R^n \to E, (f,x) \mapsto fx = \sum_ix_if_i. This map is obviously invariant under the diagonal GL_n-action on PE\times \R^n since fx = fg^{-1}gx; it is the usual identification of \R^n with the vector space E_p by means of the basis f = (f_1,\dots,f_n).

9 Connections on the tangent bundle

The tangent bundle E = TM is somewhat special since it carries another 1-form \theta besides \omega. In the moving frame language where a local frame \textup{f}= (f_1,\dots,f_n) of TM is given on an open subset M_o \subset M, any vector field Y can be written as Y= \sum \eta_i f_i. The coefficients \eta_i depend linearly on Y, and we may write \eta_i = \theta_i(Y) where the 1-forms \theta_1,\dots,\theta_n om M_o form the dual basis of (f_1,\dots,f_n), i.e. \theta_i(f_j) = \delta_{ij}. Thus

(12)Y = \sum_i \theta_i(Y)f_i

If we have a covariant derivative \nabla on TM and another vector field X, we obtain

\displaystyle  \nabla_X Y = \sum_i \left\{X(\theta_i(Y))f_i + \theta_i(Y)\nabla_Xf_i\right\} = \sum_{ij} \left\{X(\theta_i(Y))f_i + \theta_i(Y)\omega_{ji}(X)f_j\right\}

from which we derive (interchanging the roles of i and j in the second term)

\displaystyle  T(X,Y): = \nabla_XY-\nabla_YX - [X,Y] = \sum_{ij}\left(d\theta_i + \omega_{ij}\wedge\theta_j\right)(X,Y)f_i.

This tensor T : \Lambda^2TM \to TM is called Torsion tensor; letting

(13)T(.,.) = \sum\Theta_if_i

for some \Theta_i \in \Omega^2(M_o) and putting \Theta = (\Theta_1,\dots,\Theta_n)^T (called torsion form) and \theta = (\theta_1,\dots,\theta_n)^T (sometimes called soldering form), we end up with the second Cartan structure equation

(14)\Theta = d\theta + \omega \wedge \theta.

The following section explains why beneath (11) a second equation occurs for TM.

10 Affine connections

An affine frame on T_pM is a pair (f,v) where f = (f_1,\dots,f_n) is a frame of T_pM and v\in T_pM. This is acted on from the right by the affine group A_n which consists of the inhomogeneous linear transformations x\mapsto Ax+a on \R^n with A\in GL_n and a\in \R^n: we let

(15)(f,v)(A,a) = (fA,v+f^{-1}a)

where the frame f is considered as the isomorphism f :\R^n\to T_pM mapping the standard basis vector e_i\in\R^n onto f_i\in T_pM. This action turns the set AM of affine frames on TM into a A_n-principal bundle. A connection on the A_n-principal bundle AM will be called generalized affine connection. Its connection and curvature forms \hat\omega, \hat\Omega are \frak{a}_n-valued where \frak{a}_n is the Lie algebra of A_n. Since \frak{a}_n = \R^{n\times n} \oplus \R^n, the forms split accordingly into a matrix and a vector component. Now we consider the embedding \gamma: FM \to AM with \gamma(f) = (f,0). For the pull back forms on FM we have the same splitting:

(16)\gamma^*\hat\omega = \omega \oplus \theta
(17)\gamma^*\hat\Omega = \Omega \oplus \Theta

where the first term on the right takes values in \R^{n\times n}, the second on in \R^n. Moreover, the Cartan structure equations for the affine group are

(18)\begin{matrix} \Omega &=& d\omega + \omega\wedge \omega,\\ \Theta &=& d\theta + \omega\wedge\theta, \end{matrix}

using agian the splitting \frak{a}_n = \R^{n\times n}\oplus\R^n. If \theta = (\theta_1,\dots,\theta_n) has the special property (12), we call the connection affine, and \Theta equals the torsion form as introduced in the last section.

11 References

12 External links

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