7.5 Holonomy

Let $(M,g)$ be an $m$ -dimensional Riemannian manifold with an affine connection $\nabla$ . The connection naturally defines a transformation group at each tangent space $T_{p}M$ as follows.

Definition 7.3

Let $p$ be a point in $(M,g)$ and consider the set of closed loops at $p$ , $\{c(t)|0\le t\le 1,c(0)=c(1)=p\}$ . Take a vector $X\in T_{p}M$ and parallel transport $X$ along a curve $c(t)$ . After a trip along $c(t)$ , we end up with a new vector $X_c\in T_{p}M$ . Thus, the loop $c(t)$ and the connection $\nabla$ induce a linear transformation

$$P_c:T_{p}M\to T_{p}M.\text{\hspace{1em}(7.90)}$$

The set of these transformations is denoted by $H(p)$ and is called the holonomy group at $p$ .

We assume that $H(p)$ acts on $T_{p}M$ from the right, $P_{c}X=Xh$ ( $h\in H(p)$ ). In components, this becomes $P_{c}X=X^{\mu }{h}_{\mu }^{\nu }{e}_{\nu }$ , $\{e_{\nu }\}$ being the basis of $T_{p}M$ . It is easy to see that $H(p)$ is a group. The product $P_{c^{\prime }}{P}_c$ corresponds to parallel transport along $c$ first and then $c^{\prime }$ . If we write $P_d=P_{c^{\prime }}{P}_c$ , the loop $d$ is given by

$$d(t)=\{\begin{array}{ll}c(2t)& 0\le t\le \frac{1}{2}\\ c^{\prime }(2t-1)& \frac{1}{2}\le t\le 1.\end{array}\text{\hspace{1em}(7.91)}$$

The unit element corresponds to the constant map $c_p(t)=p$ ( $0\le t\le 1$ ) and the inverse of $P_c$ is given by $P_{c^{-1}}$ , where $c^{-1}(t)=c(1-t)$ . Note that $H(p)$ is a subgroup of $\mathrm{GL}(m,\mathbb{R})$ , which is the maximal holonomy group possible. $H(p)$ is trivial if and only if the Riemann tensor vanishes. In particular, if $(M,g)$ is parallelizable (see example 7.2), we can make $H(p)$ trivial.

If $M$ is (arccwise-)connected, any two points $p,q\in M$ are connected by a curve $a$ . The curve $a$ defines a map ${\tau }_a:T_{p}M\to T_{q}M$ by parallel transporting a vector in $T_{p}M$ to $T_{q}M$ along $a$ . Then the holonomy groups $H(p)$ and $H(q)$ are related by

$$H(q)={\tau }_a^{-1}H(p){\tau }_a\text{\hspace{1em}(7.92)}$$

hence $H(q)$ is isomorphic to $H(p)$ .

In general, the holonomy group is a subgroup of $\mathrm{GL}(m,\mathbb{R})$ . If $\nabla$ is a metric connection, $\nabla$ preserves the length of a vector, $g_p(P_c(X),P_c(X))=g_p(X,X)$ for $X\in T_{p}M$ . Then the holonomy group must be a subgroup of $\mathrm{SO}(m)$ if $(M,g)$ is orientable and Riemannian and $\mathrm{SO}(m-1,1)$ if it is orientable and Lorentzian.

Example 7.7

We work out the holonomy group of the Levi-Civita connection on $S^2$ with the metric $g=\mathrm{d}\theta \otimes \mathrm{d}\theta +{\sin }^2\mathrm{d}\varphi \otimes \mathrm{d}\varphi$ . The non-vanishing connection coefficients are ${\Gamma }_{\varphi \varphi }^{\theta }=-\sin \theta \cos \theta$ and ${\Gamma }_{\varphi \theta }^{\varphi }={\Gamma }_{\theta \varphi }^{\varphi }=\cot \theta$ . For simplicity, we take a vector $e_{\theta }=\partial /\partial \theta$ at a point $({\theta }_0,0)$ and parallel transport it along a circle $\theta ={\theta }_0$ , $0\le \varphi \le 2\pi$ . Let $X$ be the vector $e_{\theta }$ parallel transported along the circle. The vector $X=X^{\theta }{e}_{\theta }+X^{\varphi }{e}_{\varphi }$ satisfies

$${\partial }_{\varphi }{X}^{\theta }-\sin {\theta }_0\cos {\theta }_0{X}^{\varphi }=0\text{\hspace{1em}(7.93a)}$$

$${\partial }_{\varphi }{X}^{\varphi }+\cot {\theta }_0{X}^{\theta }=0.\text{\hspace{1em}(7.93b)}$$

Equations (7.93a) and (7.93b) represent the harmonic oscillations. Indeed if we take a $\varphi$ -derivative of (7.93a) and use (7.93b), we have

$$\frac{{\mathrm{d}}^2{X}^{\theta }}{\mathrm{d}{\varphi }^2}-\sin {\theta }_0\cos {\theta }_0\frac{\mathrm{d}{X}^{\varphi }}{\mathrm{d}\varphi }=\frac{{\mathrm{d}}^2{X}^{\theta }}{\mathrm{d}{\varphi }^2}-{\cos }^2{\theta }_0{X}^{\theta }=0.\text{\hspace{1em}(7.94)}$$

The general solution is $X^{\theta }=A\cos (C_0\varphi )+B\sin (C_0\varphi )$ , where $C_0\equiv \cos {\theta }_0$ . Since $X^{\theta }=1$ at $\varphi =0$ we have

$$X^{\theta }=\cos (C_0\varphi )X^{\varphi }=-\frac{\sin (C_0\varphi )}{\sin {\theta }_0}.$$

After parallel transport along the circle, we end up with

$$X(\varphi =2\pi )=\cos (2\pi C_0)e_{\theta }-\frac{\sin (2\pi C_0\varphi )}{\sin {\theta }_0}{e}_{\varphi }.\text{\hspace{1em}(7.95)}$$

Now the vector is rotated by $\Theta =2\pi$ cos ${\theta }_0$ , with its magnitude kept fixed. If we take a point $p\in S^2$ and a circle in $S^2$ which passes through $p$ , we can always find a coordinate system such that the circle is given by $\theta ={\theta }_0$ ( $0\le \theta <\pi$ ) and we can apply our previous calculation. The rotation angle is $-2\pi \le \Theta <2\pi$ and we find that the holonomy group at $p\in S^2$ is SO(2).

In general, $S^m$ ( $m\ge 2$ ) admits the holonomy group $\mathrm{SO}(m)$ . Product manifolds admit more restricted holonomy groups. The following example is taken from Horowitz (1986). Consider six-dimensional manifolds made of the spheres with standard metrics. Examples are $S^6$ , $S^3\times S^3$ , $S^2\times S^2\times S^2$ , $T^6=S^1\times \cdots \times S^1$ . Their holonomy groups are:

(i) $S^6:H(p)=\mathrm{SO}(6)$
(ii) $S^3\times S^3:H(p)=\mathrm{SO}(3)\times \mathrm{SO}(3).$
(iii) $S^2\times S^2\times S^2:H(p)=\mathrm{SO}(2)\times \mathrm{SO}(2)\times \mathrm{SO}(2).$
(iv) $T^6$ : $H(p)$ is trivial since the Riemann tensor vanishes.

Exercise 7.14

Show that the holonomy group of the Levi-Civita connection of the Poincaré metric given in example 7.6 is SO(2).

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