# Connections

## 1 Introduction

How can we differentiate a section $s$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}s$ in a vector bundle $E$$E$ over a manifold $M$$M$? It takes values in different vector spaces $E_p$$E_p$ for each $p\in M$$p\in M$, but differentiating involves comparing values at different points $p$$p$. This needs an extra structure on $E$$E$ which connects the different vector spaces $E_p$$E_p$, $p\in M$$p\in M$, among each other; it is therefore called connection. It can be defined in two different ways. Either it is viewed as a linear map which assigns to each vector field $X$$X$ on $M$$M$ a so called covariant derivative, a differential operator $\nabla_{\!X}$$\nabla_{\!X}$ on the space of sections on $E$$E$. Or else it is given as a parallel displacement along curves in $M$$M$: Given two points $p,q\in M$$p,q\in M$ and a curve $c$$c$ connecting these points, a connection allows to move any element of $E_p$$E_p$ to $E_q$$E_q$ along $c$$c$. Infinitesimally it is given by a distribution $\mathcal{H}$$\mathcal{H}$ on $E$$E$ or on its frame bundle $FE$$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$$E_p$ (a vector bundle) needs an extra structure called connection. The greater generality leads to a new notion: curvature. In the covariant derivative model, curvature is the commutator $[\nabla_i,\nabla_j]$$[\nabla_i,\nabla_j]$ of covariant derivatives in two coordinate directions. For ordinary derivatives $\nabla_i = \partial_i$$\nabla_i = \partial_i$ this quantity vanishes, and for covariant derivatives it is an algebraic quantity (a tensor) rather than a differential operator. In the parallel displacement model, curvature is just the non-integrability of the distribution, the tensor $[\mathcal{H},\mathcal{H}]^{\mathcal{H}^\perp}$$[\mathcal{H},\mathcal{H}]^{\mathcal{H}^\perp}$. A case of particular importance is the tangent bundle, $E = TM$$E = TM$. A connection on $TM$$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$$TM$ is sometimes called affine connection.

## 2 Covariant derivatives

A covariant derivative on a vector bundle $E$$E$ over a smooth manifold $M$$M$ is a directional derivative $\nabla$$\nabla$ for sections of $E$$E$. It can be viewed as a bilinear map $\nabla : \Gamma TM \times \Gamma E \to \Gamma E$$\nabla : \Gamma TM \times \Gamma E \to \Gamma E$, $(X,s) \mapsto \nabla_Xs$$(X,s) \mapsto \nabla_Xs$ which is a tensor (linear over $C^\infty(M)$$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}$$\begin{matrix} \nabla_{(fX)}s &=& f\nabla_Xs\,\cr \nabla_{X}(fs) &=& (Xf)s + f\nabla_Xs \,, \end{matrix}$

where $f$$f$ is a smooth function and $X$$X$ a vector field on $M$$M$ and $s$$s$ a section of $E$$E$, and where $Xf = \partial_Xf = df.X$$Xf = \partial_Xf = df.X$ is the ordinary derivative of the function $f$$f$ in the direction of $X$$X$. By these properties, $\nabla$$\nabla$ is defined locally and even pointwise regarding the first argument: For any $v\in T_pM$$v\in T_pM$ we may define $\nabla_xs := (\nabla_Xs)_p$$\nabla_xs := (\nabla_Xs)_p$ where $X$$X$ is any (local) vector field with $X_p = x$$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$$f$ we have $\partial_i\partial_jf = \partial_j\partial_if$$\partial_i\partial_jf = \partial_j\partial_if$ with $\partial_if = \frac{\partial(f\circ \phi)}{\partial u_i} = \partial_{\phi_i}f$$\partial_if = \frac{\partial(f\circ \phi)}{\partial u_i} = \partial_{\phi_i}f$, where $\phi : \R^n\to M$$\phi : \R^n\to M$ is a local diffeomorphism (local parametrization of $M$$M$) and $\phi_i := \partial_i\phi$$\phi_i := \partial_i\phi$ its $i$$i$-th partial derivative. Instead, for covariant derivatives $\nabla_i = \nabla_{\phi_i}$$\nabla_i = \nabla_{\phi_i}$ of a section $s$$s$ on a vector bundle $E$$E$, the quantity

(2)$R_{ij}s := [\nabla_i,\nabla_j]s = \nabla_i\nabla_js - \nabla_j\nabla_is$$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$$R_{ij}(fs) = fR_{ij}s$. For arbitrary vector fields $X,Y$$X,Y$ with $X \circ \phi = \sum_i\xi^i\phi_i$$X \circ \phi = \sum_i\xi^i\phi_i$ and $Y \circ \phi = \sum_j\eta^j\phi_j$$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$$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$$[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$$E$ is an $n$$n$-tuple of sections $s_1,\dots,s_n \in \Gamma E|_{M_o}$$s_1,\dots,s_n \in \Gamma E|_{M_o}$ on some open subset $M_o \subset M$$M_o \subset M$ such that for any $p\in M_o$$p\in M_o$ the vectors $s_1(p),\dots,s_n(p)$$s_1(p),\dots,s_n(p)$ form a basis of $E_p$$E_p$. Denoting by $\nabla s_j$$\nabla s_j$ the linear map $X \mapsto \nabla_Xs_j$$X \mapsto \nabla_Xs_j$, we have

(4)$\nabla s_j = \sum_i\omega_{ij} s_i$$\nabla s_j = \sum_i\omega_{ij} s_i$
for certain 1-forms $\omega_{ij}\in \Omega^1(M_o)$$\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\}$$\nabla_X\nabla_Ys_j = \sum_i \left\{X\omega_{ij}(Y)s_i + \omega_{ij}(Y)\sum_{k}\omega_{ki}(X)s_k\right\}$ and therefore (interchanging the roles of the indices $k$$k$ and $i$$i$ in the second term)
(5)$R(.,.)s_j = \sum_i \left(d\omega_{ij} + \sum_k \omega_{ik}\wedge \omega_{kj}\right)s_i$$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])$$d\omega(X,Y) = X\omega(Y) - Y\omega(X) - \omega([X,Y])$ and $(\omega\wedge\psi)(X,Y) = \omega(X)\psi(Y)-\omega(Y)\psi(X)$$(\omega\wedge\psi)(X,Y) = \omega(X)\psi(Y)-\omega(Y)\psi(X)$ for arbitrary 1-forms $\omega,\psi \in \Omega^1(M_o)$$\omega,\psi \in \Omega^1(M_o)$. On the other hand we let

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

for some 2-forms $\Omega_{ji} \in \Omega^2(M_o)$$\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}$$\Omega_{ij} = d\omega_{ij} + \sum_k \omega_{ik}\wedge \omega_{kj}$

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

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

## 5 Parallelity and connection

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

For any piecewise smooth curve $c : [a,b] \to M$$c : [a,b] \to M$ from $p$$p$ to $q$$q$ and any initial value $v \in E_p$$v \in E_p$ we have a parallel section $s$$s$ along $c$$c$ with $s(a) = v$$s(a) = v$. The mapping $\tau_c : s(a) \mapsto s(b) : E_p \to E_q$$\tau_c : s(a) \mapsto s(b) : E_p \to E_q$ is an invertible linear map called parallel transport along the curve $c$$c$. In general, parallel transport depends on the curve $c$$c$ itself, not only on the end points $p,q$$p,q$, but it is independent of the parametrization of $c$$c$. This dependence is measured by the holonomy group $\mathcal{H}ol_p$$\mathcal{H}ol_p$ at $p$$p$ which is the set of parallel transports $\tau_c$$\tau_c$ along all loops $c$$c$ at $p$$p$, i.e.piecewise smooth curves $c : [a,b] \to M$$c : [a,b] \to M$ with $c(a) = c(b) = p$$c(a) = c(b) = p$. It is known by the Ambrose-Singer theorem [Kobayashi&Nomizu1963, Theorem 8.1] that the connected component of $\mathcal{H}ol_p \subset GL(E_p)$$\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$$\tau_c^{-1}R(x,y)\tau_c$ for all curves $c : [a,b] \to M$$c : [a,b] \to M$ starting from $p$$p$ and all $x,y \in T_qM$$x,y \in T_qM$ where $q = c(b)$$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$$E$ with the linear frame bundle $FE$$FE$ whose fibre $FE_p$$FE_p$ over $p\in M$$p\in M$ is the set of all frames (bases) of the vector space $E_p$$E_p$. Then parallelity of a frame $(s_1,\dots,s_n)$$(s_1,\dots,s_n)$, i.e. parallelity of all sections $s_i$$s_i$ in this frame, is expressed by a distribution on $FE$$FE$ which is also called $\mathcal H$$\mathcal H$. Together with the the "vertical space" $\mathcal V_f$$\mathcal V_f$, the tangent space $T_f(FE_p)$$T_f(FE_p)$ of the fibre through $f$$f$, it yields a direct decomposition $T_fFE = \mathcal{H}_f \oplus \mathcal{V}_f$$T_fFE = \mathcal{H}_f \oplus \mathcal{V}_f$ and hence we have projections $\pi_{\mathcal H}$$\pi_{\mathcal H}$ and $\pi_{\mathcal V}$$\pi_{\mathcal V}$ of $TFE$$TFE$ onto the two subbundles. $FE$$FE$ is a principal fibre bundle for the group $G = GL_n$$G = GL_n$, i.e.the fibres are the orbits of a free action of $G$$G$ from the right given by $(f,g) \mapsto fg$$(f,g) \mapsto fg$ where $f = (f_1,\dots,f_n)$$f = (f_1,\dots,f_n)$ is a frame and $g = (g_{ij})$$g = (g_{ij})$ a matrix and where $fg$$fg$ is the line with $j$$j$-th component $\sum_i f_ig_{ij}$$\sum_i f_ig_{ij}$. Fixing $f\in FE_p$$f\in FE_p$, the action $\phi_f : G \to FE_p$$\phi_f : G \to FE_p$, $g\mapsto fg$$g\mapsto fg$ is a diffeomorphism which is equivariant with respect to right translation, and the left invariant vector fields on $G$$G$ are turned by $\phi_*$$\phi_*$ into vector fields on $FE$$FE$ tangent to the fibres, so called fundamental vector fields. In particular, the vertical space $\mathcal{V}_f$$\mathcal{V}_f$ is canonically isomorphic to the Lie algebra $\frak{g}$$\frak{g}$ of $G$$G$, via the infinitesimal action $(\phi_f)_* = d(\phi_f)_e$$(\phi_f)_* = d(\phi_f)_e$. Using this identification, the projection $\pi_{\mathcal V}$$\pi_{\mathcal V}$ is a linear form on $TFE$$TFE$ with values in $\frak{g}$$\frak{g}$; it will be called (global) connection form $\omega$$\omega$. The form $\omega$$\omega$ in Section 4 will be better called $\omega_{\textup{f}}$$\omega_{\textup{f}}$ since it depends on a moving frame $\textup{f}= (s_1,\dots,s_n) : M_o \to FE$$\textup{f}= (s_1,\dots,s_n) : M_o \to FE$. We have

(9)$\omega_{\textup{f}} = \textup{f}^*\omega.$$\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$$d\omega + \omega\wedge\omega = \Omega$

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

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

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

## 8 Connections on general principal bundles

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

## 9 Connections on the tangent bundle

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

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

If we have a covariant derivative $\nabla$$\nabla$ on $TM$$TM$ and another vector field $X$$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$$i$ and $j$$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$$T : \Lambda^2TM \to TM$ is called Torsion tensor; letting

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

for some $\Theta_i \in \Omega^2(M_o)$$\Theta_i \in \Omega^2(M_o)$ and putting $\Theta = (\Theta_1,\dots,\Theta_n)^T$$\Theta = (\Theta_1,\dots,\Theta_n)^T$ (called torsion form) and $\theta = (\theta_1,\dots,\theta_n)^T$$\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.$$\Theta = d\theta + \omega \wedge \theta.$

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

## 10 Affine connections

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

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

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

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

where the first term on the right takes values in $\R^{n\times n}$$\R^{n\times n}$, the second on in $\R^n$$\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}$$\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$$\frak{a}_n = \R^{n\times n}\oplus\R^n$. If $\theta = (\theta_1,\dots,\theta_n)$$\theta = (\theta_1,\dots,\theta_n)$ has the special property (12), we call the connection affine, and $\Theta$$\Theta$ equals the torsion form as introduced in the last section.