Riemann, Ambrose, Singer, Berger

The last GR problem displays a deep correlation between the Riemann Tensor, i.e. the Curvature of a Riemannian manifold and the change of a vector field if it is parallel transported around a loop.

The stuff is easy to picture. Take a tangent vector moving around a loop in flat space. When you arrive at the starting point it will still point in the same direction . To change direction something needs to happen along the route. Thats curvature.

Imagine the vector as a ship. A strange ship because it has no rudder… you should not be able to change direction “by hand”. Now you start sailing around the globe. The globe can be seen as a curved manifold (\cong S^2)-all land flooded. Consider 2 different ways.

First, simply sail around the equator (say starting at Somalia, heading for Indonesia). When you arrive at the start you still see the coastline of Indonesia (if it wasnt´ submerged ). That should not be surprising, because what you essentially did was sailing around a flat manifold, the equator (\cong S^1).

Now lets start at Somalia again. This time heading for the north pole. We sail till we ve circled half of the globe. We are now at the equator , but at the opposite side of the earth (somewhere in the pacific ocean ). We close the loop by sailing back along the equator. Our strange ship doesn´t allow us to turn around and so, when we arrive at Somalia, we look at the south pole. A 180 degree rotation.

Different ways would have yield different rotations :

Lets chip in some mathematical notation. Take a smooth  manifold (connected is improtant here, because we want to close loops). At the starting point  choose a tangent vector V. Obviously from the tangent space T_pM at that point. After we terminate our journey we obtain a rotated vector V` from the same tangent space. We can map the two vectors into each other by a GL(n) operation . Therefore we have a group operation on the tangent bundle, generated by parallel transport along any closed loop on the manifold. Its called the Holonomy group.

Parallel transporting around an infinitesimal parallelogram complies with taking the covariant derivative along a vector field \partial_{\mu} , after that along another field \partial_{\nu} and reversing the procedure to come back where we started. Lets calculate this in a map:

\left[D_{\partial_{\mu}},D_{\partial_{\nu}}\right](\partial_{\tau})=\, D_{\partial_{\mu}}(D_{\partial_{\nu}}(\partial_{\tau}))-D_{\partial_{\nu}}(D_{\partial_{\mu}}(\partial_{\tau}))\newline \hspace*{27mm}=\partial_{\mu}\Gamma^{\sigma}_{\nu\tau}\partial_{\sigma}+\Gamma^{\sigma}_{\nu\tau}\Gamma^{\tau}_{\mu\sigma}\partial_{\tau}-(\mu\leftrightarrow \nu)\newline \hspace*{27mm}=\left[\partial_{\mu}\Gamma^{\tau}_{\nu\tau}+\Gamma^{\sigma}_{\nu\tau}\Gamma^{\tau}_{\mu\sigma} -(\mu\leftrightarrow \nu) \right]\partial_{\tau} \newline \hspace*{27mm} = R^{\sigma}_{\tau\mu\nu}

It follows the statement of the Ambrose-Singer Theorem:

“The Lie Algebra of the Holonomy group (at some point p) \mathcal{H}ol_p(\omega) is generated by the values of the curvature tensor R(X,Y)(p) (at that point), as X,Y vary over all possible vector fields (which flows describe closed loops). ”

The \omega stands for the connection.

It is obvious that the Holonomy group of a 1 dimensional manifold is trivial,since \left[\partial_{\mu},\partial_{\nu}\right]=0 . But what can be said about general (semi-)Riemannian manifolds? Parallel transport preserves the metric. Therefore the induced linear transformations on tangent space does it. Hence, the Holonomy group is a subgroup of the isometric transformations. Thats O(n) in the case of a Riemannian manifold. If the manifold is orientable, vector flipping will not occur, so its SO(n)  (Take a look at the tangent frame in the gif and you will agree). In the case of GR it becomes the connected component of the Lorentz group SO(p,q).

The  classification of holonomy groups was done by Berger and is the subject of another theorem. It also includes complex manifolds like Calabi-Yaus, which are popular candidates if string theorists want to compactify a bunch of extra dimensions.

It should be mentioned that the concept of holonomy plays an important role in geometric gauge theory and is a cornerstone of loop quantum gravity.

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