Local formulas

From now on, we assume that $U \subset M$ is an open chart domain over which the tangent and cotangent bundles are trivial. Local coordinates are functions $x^1,\dots,x^n \in \Coo(U)$, and we denote $\del_j := \del/\del x^j \in \gX(M)\bigr\vert _U$ for the local basis of vector fields; by definition, their Lie brackets vanish: $[\del_i, \del_j] = 0$. We define the Christoffel symbols $\Ga_{ij}^k \in \Coo(U)$ by

$$\nb_{\del_i} \del_j =: \Ga_{ij}^k \del_k, \words{or} \nb \del_j =: \Ga_{ij}^k dx^i \ox \del_k.$$

The explicit expression () for the Levi-Civita connection reduces to a local formula over $U$, namely

$$\Ga_{ij}^k := \half g^{kl}(\del_i g_{jl} + \del_j g_{il} - \del_l g_{ij}); \quad\word{here} [g^{rs}] = [g_{ij}]^{-1}.$$ (10)

Notice that $\Ga_{ji}^k = \Ga_{ij}^k$; this is beacuse of torsion freedom.

Dually, the coefficients of the Levi-Civita connection on $1$-forms are $-\Ga_{ij}^k$ (note the change of sign):

$$\nb_{\del_i}(dx^k) = - \Ga_{ij}^k dx^j, \words{or} \nb(dx^k) = - \Ga_{ij}^k dx^i \ox dx^j.$$

Since the Riemannian metric gives a concept of (fibrewise) orthogonality on the tangent and cotangent bundles, we can select local orthonormal bases:
$$\begin{align*} \{E_1,\dots,E_n\} &\word{for} \gX(M)\bigr\vert _U = \Ga(U, TM): ... ..._U = \Ga(U,T^*M):\quad g(\th^{\al}, \th^{\bt}) = \delta^{\al\bt}. \end{align*}$$
We rewrite the Christoffel symbols in these local bases:

$$\nb E_\al =: \Tilde{\Ga}_{i\al}^\bt dx^i \ox E_{\bt}, \qq... ... \nb \th^{\bt} = - \Tilde{\Ga}_{i\al}^\bt dx^i \ox \th^\al.$$

Metric compatibility means that, for each fixed $i$, the $\Tilde{\Ga}_{i\8}^\8$ are skewsymmetric matrices:

$$\Tilde{\Ga}_{i\al}^\bt + \Tilde{\Ga}_{i\bt}^\al = - g(\nb_... ...th^\bt,\nb_{\del_i}\th^\al) = - \del_i(\delta^{\al\bt}) = 0.$$

Thus $\Tilde{\Ga}$ lies in $\sA^1(U, \gso(T^*M)) \isom \sA^1(U) \ox_\bR \gso(\bR^n)$.

Before discussing existence, let us look first at local formulas. We thus write `` $\nb = d - \Tilde{\Ga}$ locally'' for the Levi-Civita connection, with an implicit choice of local orthonormal bases of $1$-forms. We recall that there are isomorphisms of Lie algebras

$$\dot{\mu} \: \gso(T_x^*M) \to Q(\La^2 T_x^*M) \equiv \gspin(T_x^*M)$$

with the property that $\ad(\dot\mu(A)) = A$ for $A \in \gso(T_x^*M)$; in other words, $[\dot\mu(A), v] = Av$ for $v \in T_x^*M$ --this is a commutator for the Clifford product in $\Cl(T_x^*M, g_x)$. On the chart domain $U$, we can apply $\dot{\mu}$ to $\Tilde{\Ga}$ fibrewise; this means that

$$[\dot\mu(\Tilde\Ga), c(\al)]= c(\Tilde\Ga \al)$$

for $\al \in \sA^1(M)$ with support in $U$, $\Tilde\Ga \in \Ga(U,\End T^*M)$ is mapped to $\dot\mu(\Tilde\Ga) \in \Ga(U,\End S)$, and $c(\al)$ again denotes the Clifford action action of $\al$ on $\sS\bigr\vert _U = \Ga(U,S)$.

In this way we get the local expression of a connection,

$$\nb^S := d - \dot{\mu}(\Tilde{\Ga}), \words{acting on} \sS\bigr\vert _U.$$ (11)

Suppose we take $\al \in \sA^1(M)$ with support in $U$, and $\psi \in \sS\bigr\vert _U$. Then

Thus $\nb^S := d - \dot{\mu}(\Ga)$ provides a local solution to the existence of $\nb^S \: \sS \to \sA^1(M) \ox_{\sA} \sS$ satisfying the Leibniz rule:

$$\nb^S(c(\al)\psi) = c(\nb \al) \psi + c(\al) \nb^S\psi.$$

Physicists like to write $\ga^{\al} := c(\th^{\al})$ for a given local orthonormal basis of $\sA^1(M)$ --so that the $\ga^\al$ are fixed matrices. For convenience, we also write $\ga_\bt = \delta_{\al\bt} \ga^\al$ also (in the Euclidean signature, which we are always using here); in other words, $\ga_\bt = \ga^\bt$ but with its index lowered for use with the Einstein summation convention. Thus the Clifford relations are just

$$\ga^\al \ga^\bt + \ga^\bt \ga^\al = 2\delta^{\al\bt}, \word{for} \al,\bt = 1,\dots,n.$$

The formula () for $\dot\mu$ can now be rewritten as

$$\dot\mu(\Tilde\Ga) = - \quarter \Tilde\Ga_{\8\al}^\bt \ga^\al \ga_\bt.$$

A more sensible notation arrives by introducing matrix-valued functions $\om_1,\dots,\om_n \in \Ga(U, \End T^*M)$ as follows:

$$\om_i := - \quarter \Tilde\Ga_{i\al}^\bt \ga^\al \ga_\bt.$$

Let us look at the calculation () again, after contracting with a vectorfield $X$. We get
$$\begin{align*}[\nb^S_X, c(\al)] \psi &= [\sL_X - X^i \dot\mu(\om_i), c(\al)] \... ...\sL_X \al) - X^i c(\om_i \al) \bigr) \psi = c(\nb_X \al) \psi. \end{align*}$$
Thus the local coefficients of $\nb^S_X$ are $-\quarter X^i \Tilde{\Ga}_{i\al}^\bt \ga^\al \ga_\bt$, for $X \in \gX(M)$.

Now suppose $S$ comes from a spin structure on $M$. Since $C(\psi a) = (C\psi)\bar a$ for $a \in \sA = \Coo(M)$, the operator $C$ acts locally (as a field of antilinear conjugations $C_x\: S_x \to S_x$); and since $C(b) = \chi(\bar b)C$ for $b \in B$, we get, for $\al,\bt = 1,\dots,n$:

$$C \ga^\al \ga^\bt = C c(\th^\al) c(\th^\bt) = c(\th^\al) c(\th^\bt) C = \ga^\al \ga^\bt C.$$

Thus $\nb^S_X C - C \nb^S_X$ vanishes over $U$, provided $X\bigr\vert _U$ is real.

Suppose that $\nb'$ is another local connection defined on $\sS\bigr\vert _U$ and satisfying the Leibniz rule () there. Then

$$\nb' - \nb^S = \bt \in \sA^1(U, \End S)$$

and $c(\ka) \bt\psi = \bt c(\ka)\psi$ for all $\ka \in \sB\bigr\vert _U$. Thus $\bt_x$ is a scalar matrix in $\End(S_x)$, for each $x \in U$. To fix $\bt$, we ask that both $\nb'$ and $\nb^S$ be Hermitian connections; this entails that each $\bt_x$ is skew-hermitian:

$$\<\bt_x\phi_x\vert\psi_x> + \<\phi_x\vert\bt_x\psi_x> = 0 \word{for} x \in U,$$

so this scalar is purely imaginary. On the other hand, if $\nb'_X$, like $\nb^S_X$, commutes with $C$ whenever the coefficients $X^i$ of $X$ are real functions, then this scalar must be purely real. Therefore, $\bt = 0$.

If $\nb$ is any connection on an $\sA$-module $\sE = \Ga(M, E)$, then

$$\nb^2(fs) = \nb(df \ox s + f \nb s) = \bigl( d(df) \ox s ... ... \bigr) + \bigl( df \nb s + f \nb^2 s \bigr) = f \nb^2 s,$$

for $f \in \sA$, so that $\nb^2$ is tensorial: $\nb s = R s$ for a certain $2$-form $R \in \sA^2(M, \End E)$, the curvature of $\nb$. For the Levi-Civita connection, a local calculation gives

$$\nb^2 \al = (d - \Tilde\Ga)(d\al - \Tilde\Ga \al) = -d(\T... ...lde\Ga \al) = (-d\Tilde\Ga + \Tilde\Ga \w \Tilde\Ga) \al,$$

which yields the local expression for the Riemannian curvature tensor:

$$R\bigr\vert _U = -d\Tilde\Ga + \Tilde\Ga \w \Tilde\Ga \in \sA^2(U, \gso(T^*M)).$$

Likewise, the curvature $R^S$ of the spin connection is locally given by

$$\dot\mu(R) = - d\dot\mu(\Tilde\Ga) + \dot\mu(\Tilde\Ga) \w \dot\mu(\Tilde\Ga) \in \sA^2(U, \End S).$$

One can check these formulas to get more familiar expressions by computing $R(X,Y) = \iota_Y\iota_X R$ and likewise $R^S(X,Y)$, for $X,Y \in \gX(M)$.