History of Connections

I haven’t blogged for our seminar in quite a while and thought I might fix this with this short post.

We are now discussing connections on Riemannian manifolds — more accurately, affine connections. Such an object allows us to differentiate a vector field, $Y$, along another vector field, $X$, at any point $p \in M$.  The result of this process is typically notated as $\nabla_X\,Y$, and, however its defined, it must enjoy three properties:

1. $\nabla_{fX+gY}Z = f\nabla_XZ + g\nabla_YZ$
2. $\nabla_X(fY) = (X(f))Y + f\nabla_XY$
3. $\nabla_X(Y+Z) = \nabla_XY + \nabla_XZ$

These conditions are motivated by what happens in an easier-to-understand setting, like $\mathbb{R}^n$, where one can differentiate vector fields along other vector fields in an obvious way.

Given a Riemannian metric $\langle, \rangle$ on $M$, it turns out there always exists a special affine connection, one referred to as the Levi-Civita connection on $M$.  It is defined by being both (1) compatible with the metric and (2) symmetric.  Compatibility means that for all vector fields $X, Y,$ and $Z$ one has

$\displaystyle \nabla_X\langle Y, Z \rangle = \langle \nabla_XY,Z\rangle + \langle Y, \nabla_XZ\rangle$

and symmetry means that

$\displaystyle \nabla_XY - \nabla_YX =[X, Y].$

This is all well and good, but it can nonetheless feel too detached from intuition.  For instance, one might wonder why we use these properties to define an affine connection as opposed to other ones.  Similarly, why would we want compatibility to hold and why would we want symmetry?

For answers to these and other, related questions, you should consult this Masters Thesis.  Its from 2008, and was written by Kamielle Freeman; the title of the manuscript is “A Historical Overview of Connections in Geometry.”  It turns out that there are lots of ways to approach and motivate the idea of a connection, and during the late 1800s and early 1900s, various mathematicians did.  One main idea can be (roughly) summarized as follows:

parallel transport $\iff$ affine connection

This idea of “parallel transport” is what allowed mathematicians to think about how to “connect” two different tangent spaces attached to the same manifold.

Q: How do I think of this vector $\vec{v} \in T_pM$ as a vector in $T_qM$?

A: Parallel transport the vector $\vec{v} \in T_pM$ to a unique vector $w \in T_qM$

The ability to move $\vec{v}$ to a new vector in a different tangent space implies the ability to differentiate vector fields along other vector fields.  Conversely, the ability to differentiate vector fields along other vector fields implies ability to “parallel transport.”

A Brief Note on Tensors

We’ve known for a few weeks now that vector fields are sections of a particular vector bundle over our manifold $M$, i.e. the tangent bundle $TM$.  As we’ve (lightly) discussed, though, there are other vector spaces one can bundle over $M$, and there exist smooth sections of these bundles, too.  For instance, the co-tangent bundle, $T^*M$, has as its sections differential 1-forms.

tensor over $M$ is a section of a more general kind of vector bundle (which is why this is also some times called a tensor field).  The vector bundles that give rise to tensor fields can be built out of interesting combinations of the two bundles $TM$ and $T^*M$.  Once we have an affine connection $\nabla$ on a Riemannian manifold, we can use it to build a connection for $T^*M$, i.e. we can differentiate co-vector fields along vector fields.  This process can be continued further, in fact; we can use $\nabla$ (and the underlying metric $\langle\, , \,\rangle = g$) to  differentiate any tensor field along a vector field.