What is a connection? (Intro)
I can answer that in a number of ways and I will do so in hand-wavily. After all, this is the introductory post. A connection is:
(1) a differential operator of sections of a vector bundle E along vector fields. In turn, its output will be another section of E. This is the generalisation of the directional derivative of vector fields.
(2) a means of transporting vectors from one fibre of E to another along a curve on the base manifold M. Fibres of a vector bundle cannot be identified with each other in a canonical way so a connection is a choice of that identification.
(3) a choice of a horizontal subbundle that decomposes the tangent bundle TE of E. There are many ways of decomposing vector spaces. A connection is a choice of decomposing the tangent space of a vector bundle. (1) embodies the infinitesimal outlook and axiomatic definition of connections. (2) embodies the local outlook and describes connections in a more geometric, intuitive way. (3) is a special case of Ehresmann connections on fibre bundles. The beauty lies in the fact that these avatars are all equivalent! Over the next few posts, we will discuss each of them and their relationships with each other. Now at this point, one might ask what are connections used for? Hopefully, one can see how they are useful in their own right when we discuss (1)-(3). In terms of their applications, I will share how they have been useful to me during the trembling two years of my mathematical existence.
1) Geometric flows: Connections allow one to define the notion of curvature. In the case of plane curves, the curvature measures how rapidly a curve pulls away from its tangent line. It is a geometric invariant that is captured by the acceleration vector, or rather, second derivatives. Likewise, from the view point of connections as differential operators, the curvature is just the square of a connection. Curvature itself has made notable appearances in various geometric flows, such as the Ricci flow and Mean curvature flow. These flows, which are differential equations involving a time-dependent metric and curvature, allows one to deform the geometry of their favourite manifold over time. 2) Choosing geometric structures on vector bundles: In complex geometry, smooth connections allow one to define a holomorphic structure on a smooth vector bundle, which comprises of an open cover of a base manifold, holomorphic sections and holomorphic transition maps. Every choice of a holomorphic structure on a smooth vector bundle defines a holomorphic vector bundle. But, how do we choose these structures? Recall that the exterior derivative can be decomposed into its holomorphic and anti-holomorphic components. We then define a function to be holomorphic if and only if it vanishes under the antiholomorphic differential. In the same way, a smooth connection, as a differential operator, can be broken into its holomorphic and anti-holomorphic components. This gives us a way of picking out holomorphic sections because we can just define a section to be holomorphic if and only if they vanish under the anti-holomorphic differential. The upshot? One can describe the space of isomorphism classes of holomorphic vector bundles by parametrising anti-holomorphic components of smooth connections. The idea of relating geometric structures to connections is especially crucial to the proof of non-abelian Hodge theory. In particular, by multiplying a connection by a parameter t in [0, 1], we can vary from one geometric structure (at t = 0) to another (at t = 1). Throughout this blog, we will build the story of connections, piece by piece, and with as many pictures as possible!