Affine connection, metric and parallel transport and mutual interdependence

Question

I am eternally confused even after repeated learning about the mutual independence between affine connections and the metric tensor and parallel transport. Given any one of them, can I recover the rest?? I know that given any smooth manifold, there are infinitely many affine connections defined. What exactly are the requirements for the existence of each of the three mentioned above?? Wikipedia says that connections and parallel transport can each help define the other. I know this is a fairly elementary question, but I am looking for an answer which will help me recall it again without confusion.

Lastly, is the Lie bracket a necessity or sufficient for defining any of these? Hope I haven't crammed too many questions into one.

Answer 1

Well, if we just pull all the things together, the pile may look confusing. A system could help to bring in an order. There may be no best way to do this, and everyone would organize these concepts to their own taste and experience. I will try to share my way of seeing these concepts. Please consider these notes as an informal overview. The precise definitions and statements you can find, for instance, in the references to this answer.

A smooth manifold possesses something which we call a smooth structure. This structure gives rise to vector fields, differential forms, tensor fields, the Lie derivatives, the exterior derivative on forms, - and all these things age given to us naturally, that is they don't require any choice whatsoever to be defined. Following this direction, one comes up with the idea of natural (vector) bundles and natural differential operators on manifolds.

Lie derivatives and the exterior derivative are some examples of natural differential operators. Tensor bundles, jet bundles give us examples of natural bundles.

In the real life applications, we deal with manifolds equipped with an additional structure, which is usually a choice of some section of a natural vector bundle, or a differential operator between natural bundles. These choices are made with certain nice behavior of the resulting structure in mind.

For instance, we choose a positive-definite symmetric covariant tensor of valence 2 and call it a Riemannian metric. A symplectic structure is a choice of a non-degenerate antisymmetric covariant tensor of valence 2. There are many other examples.

When we make choice, we usually expect the abundance of options to choose from. This is the case with Riemannian metrics, connections, but symplectic structures are only available on even-dimensional manifolds (trivially). There are more sophisticated examples, of course.

If we choose a connection on a manifold, there may not much to say about it, but one thing is there: a parallel transport. In fact, choosing a connection or a parallel transport are equivalent to each other (the standard exercise).

As I mentioned in the beginning, there are natural differential operators on manifolds, namely Lie derivatives and the exterior derivative. The issue with the exterior derivative is it is only defined on antisymmetric covariant tensors (differential forms). The Lie derivative $\mathfrak{L}_X$, on the other hand is defined on all tensor fields, but it "is not tensorial in the $X$ slot", so the iterated application of it is somewhat complicated. And yes, the iterated application of the exterior derivative vanishes: $\mathrm{d} \circ \mathrm{d} = 0$.

Connections are tensorial in the $X$ slot, so we can more easily iterate their action (at least, on tensor bundles), and this gives us, among other things, a notion of the curvature operator of the connection. I recommend this answer, if you want to make these things precise.

The Koszul formula (see P.Petersen, Riemannian geometry, Chapter 2) can be interpreted as the calculation of the Levi-Civita connection in terms of the Lie derivative, the exterior derivative, and the Riemannian metric:

$$ 2 g( \nabla_Y X, Z) = (\mathfrak{L}_X g) (Y,Z) + ( \mathrm{d} \theta_X ) (Y,Z) $$

where $\theta_X (Y) = g(X,Y) $, or, in other terms, $\theta_X = X^{\flat}$.

To summarize, given a Riemannian metric $g$ on a (smooth) manifold, you get a uniquely defined (by a formula!) connection, with respect to which your metric is parallel: $\nabla g = 0$. This gives you the Riemannian tensor calculus, and lots of other useful things, known collectively as Riemannian geometry.

Of course, one can do similar things with other kinds of structure, and develop other geometric disciplines.

Answer 2

Here is list of some interactions between these three concepts.

The notion of Parallel transort requires a connection.

A general connection does not define a metric.

Given a Riemannian metric, there is a unique connection on the tangent bundle (Levi-Civita) which is compatible with the metric.

In more detail

A connection $\nabla$ is can be defined on any vector bundle $E\to X$ (not just the tangent bundle). Really, the concept of a connection is the same as parallel transport. A connection gives you a special horizontal distribution of the associated principal $GL_n$-bundle (general linear group on $\mathbb{R}^n$ for a rank $n$-bundle). This distribution in fact prescribes what it means for a section to be parallel transported along a curve $\gamma$ in $X$.

A connection can also be defined as a certain matrix valued 1-form ${\cal A}$ on $X$. One can show that this definition is equivalent to the one involving a choice of Horizontal distribution. The curvature of the connection is defined in terms of the Lie bracket (commutator) $${\cal F}=d{\cal A}+[{\cal A},{\cal A}]$$

Now a Riemannian metric $g$ lives on the tangent bundle $TX\to X$. It does the thing we expect, which is to allow us to measure angles and distances locally. It is a theorem that there exists a unique torsion-free connection $\nabla_{g}$, which is compatible with $g$. This connection is called the Levi-Civita connection.

Some examples

Given a Riemannian manifold $(X,g)$, the Christoffel symbols $\Gamma$ are the tensor components for the Levi-Civita connection on the tangent bundle. The curvature of this connection is the Riemann curvature tensor.

Consider the trivial vector bundle $\mathbb{R}\times S^1$. The classical closed-non exact form $$\omega=\frac{y}{x^2+y^2}dx-\frac{x}{x^2+y^2}dy$$ defnes a flat connection on this trivial bundle. The parallel transport associated with this connection gives a lift into the universal cover of $S^1$.

Edit

Ok, so here is some more detail.

First, there is a more general concept of metric. It is the following:

Let $\pi:E\to X$ be a vector bundle. Suppose there is a bundle map $g:E\otimes E\to \mathbb{R}\times X$ whose restriction to each fiber defines an inner product $$g:\pi^{-1}(p)\otimes \pi^{-1}(p)\to \mathbb{R}\;.$$ Then we say that $g$ defines a metric on $E$.

The fundamental theorem of Riemannian geometry works (in part) in this more general context. That is, if $\pi:E\to X$ is equipped with a metric $g$. Then there is a connection $\nabla$ on $E$ which is compatible with $g$ (see for example M. Nakahara "Geometry, topology and physics theorem 10.33).

However, there are possibly many connections on $E$ which have nothing to do with $g$. I have only said that there is one which is compatible.

Turning to the definition of connection and curvature:

A connection on a vector bundle $\pi:E\to X$ is equivalently one of the following:

1. A Horizontal distribution $HP$ of the associated frame bundle $P\to X$ which is "equivariant" with respect to the action of $GL_n$ on the fibers. That is, we have $$gH_{p}P=H_{pg}P,\ \ g\in GL_n$$

2. A Lie algebra valued 1-form ${\cal A}\in \Omega^1(E;M_{n})$ (M_{n} is all $n\times n$ matrices: the Lie algebra of $GL_n$) (satisfying a few additional properties).

The equivalence between the two definitions essentially uses the fact that such a horizontal distribution $HP$ has a complement $VP$ which is fiberwise isomorphic to $M_n$ and $P$ decomposes as $P=HP\oplus VP$. One needs to show that projection onto $VP$ defines a Lie algebra valued (matrix valued) 1-form.

Given the Lie algebra definition, we can define the curvature $${\cal F}=d{\cal A}+[{\cal A},{\cal A}]\;.$$ The bracket is defined as a combination of the wedge product and bracket. If you write out a matrix valued form ${\cal A}$ as a real $n\times n$ matrix, it will look like a matrix with components given by 1-forms. Then the bracket is defined by doing the usual commutator of ${\cal A}$ with itself, but you use the wedge product to multiply the entries.

The exterior derivative just takes the derivative of each entry.

Edit 2

A bit more on what a connection "does".

With regards to your intuition on connection: your not that far off. I would be more precise though and say that a connection tells how relate the value of a section in one fiber to that in another in order to make sense of the derivative as a sort of "limit of the difference quotient" which a-priori won't make sense.

A metric, on the other hand, does not relate the values of sections in different fibers. What it does to is tell you is the angle between the value of two sections (vector quantities) in every given fiber. It also says that this angle varies smoothly as we vary the base point.