Can we bypass connection?

Question

I am new to differential geometry. It is surprising to find that the linear connection is not a tensor, namely, not coordinate-independent.

Can we bypass this ugly object? Only intrinsic quantities should appear in a textbook.

Answer 1

Connections are not tensors, but that does not mean they are not coordinate-independent objects! A linear connection is a map sending two vector fields $X,Y$ to another vector field $\nabla_X Y$ which satisfies the rules $\nabla_{fX+Z} Y = f \nabla_X Y + \nabla_Z Y$ and $\nabla_X (fY + Z) = f\nabla_X Y + (\nabla_X f)Y + \nabla_X Z$. This is an abstract definition that makes no reference whatsoever to coordinates.

The connection coefficients or Christoffel symbols $\Gamma^i_{jk} = (\nabla_j\partial_k)^i$ describe how the connection acts on a given coordinate basis. This object $\Gamma$ is not a tensor precisely due to the second rule above: it satisfies a product rule rather than full bilinearity. (Indeed this is necessary if we want something that behaves like a derivative - tensors act pointwise and thus cannot distinguish constants from non-constants!) This does not mean it's not intrinsic - it just means that its components in different coordinate systems are not related in the same way that those of tensors are.

Answer 2

In contrast to what is written in the question and some of the comments, a connection is independent of coordinates.

The intrinsic way to define connections is the following. For simplicity, we treat only connections on the tangent bundle, even though a similar definition can be applied for any vector bundle. Let $M$ be a smooth manifold. Let $TM$ denote the tangent bundle to $M$, let $\Gamma(TM)$ denote the space of vector fields on $M$, and let $\mathrm{end}(TM)$ denote the space of vector bundle morphisms from $TM$ to itself. A connection on $M$ is a map$$\nabla:\Gamma(TM)\to\mathrm{end}(TM),$$which is additive and satisfies the Leibniz rule. Additivity should be clear enough, and the Leibniz rule means that we have $$\nabla fX=df\cdot X+f\nabla X,$$for any smooth function $f$ and a vector field $X$.

It is customary to use the notation $\nabla_YX$ for $\nabla(X)(Y)$. Using this notation, additivity means $$\nabla_Y(X_1+X_2)=\nabla_YX_1+\nabla_YX_2,$$while the Leibniz rule takes the form$$\nabla_YfX=df(Y)\cdot X+f\nabla_YX.$$As written above, this is perfectly intrinsic. Now, in some schools this intrinsic approach is neglected, and connections are introduced only via the Christoffel symbols. Naturally, the Christoffel symbols depend on coordinates, but this actually means nothing.

Answer 3

As mentioned in a previous answer, connections are quite intrinsic. I will take the slightly more pedestrian view that a connection is a collection of maps from type $(k,l)$ tensor fields to type $(k,l+1)$ fields that is linear, satisfies a Leibniz rule, commutes with contraction, agrees with the differential for smooth functions and is usually assumed to be torsion free in the sense that $\nabla_a\nabla_bf = \nabla_b \nabla_a f$, where small latin indices are used in the abstract index sense (apologies to those who use precisely the opposite), so that second covariant derivatives of smooth functions are symmetric.

Now suppose we have two connections $\nabla$ and $\tilde{\nabla}$. We already know that $(\nabla - \tilde \nabla) f = 0$ for any smooth function. Now suppose we have a one form $\omega_a$. Then we can compute that $$ (\nabla_b - \tilde \nabla_b)(f\omega_a) = f[(\nabla_b - \tilde \nabla_b)(\omega_a)] $$ as a result of the Leibniz rule. This means that $\omega_a \to (\nabla_b - \tilde \nabla_b)(\omega_a)$ is linear over $C^\infty(M)$, and thus is characterized by some type $(1,2) $ tensor $C^c_{ab}$ such that $(\nabla_b - \tilde \nabla_b)(\omega_a) = \omega_c C^c_{ab}$. The difference between the two actiong on arbitrary fields can now be expressed by various signed sums of contractions with $C^c_{ab}$

Now, there are particular connections $\nabla$ that we want to be able to compute with, e.g. Levi-Civita, which are defined in terms of intrinsic properties of $M$. But the only connections we really know how to write down easily are the flat connections $\tilde\nabla$ associated to a fixed coordinate chart. In this case, we often write $C^c_{ab} = \Gamma^c_{ab}$, which are the Christoffel symbols. Note that the "ugliness" of these symbols has to do with the fact that the flat connections are not coordinate independent, thus for different coordinate charts one generally has a different correction tensor $C^c_{ab}$, which leads to the saying which rather peeved a physics professor of mine that the Christoffel symbols are not a tensor.

Additionally, it seems rather strange to say that we should never present anything extrinsic. I suppose it is possible that everything could be derived from purely intrinsic computation, but that seems strange. In particular the use of differential geometry has many practical uses where we really would like to be able to numerically compute,say, the path of a geodesic.