(Disclaimer: I am writing based on what I remember from one seminar that I heard given by Alain goriely, and so claim responsibilities for all inaccuracies!)
Basic differential geometry has been applied to the problem of protein folding and dynamics of protein in an interesting way (and this is fairly new work from what I gathered). Here I give one simple example:
A little bit of remedial biology in case we forgot: a protein molecule is formed by one or more polypeptide chains. Each chain is made of building blocks called amino acids joined together by peptide bonds. As its name suggests, a polypeptide chain is just a long string of amino acids chained end to end. What determines the shape of protein molecules is the individual amino acids. Roughly speaking, each building block (amino acid) is formed by a backbone (something common to all amino acids), together with one or more things than hangs off the backbone. The backbone gives the initial chain like structure of the polypeptide chain. The interaction between the things hanging off the backbone, and between the things and the surrounding environment, is what drives the dynamical folding of the protein giving its final shape.
For traditional protein dynamics, or for traditional storage of protein structures, what they do is they take the numerically computed (or experimentally observed) protein structure, and define a map $\pi$. The map $\pi(n)$ roughly gives the (relative) spatial position of the $n$th amino acid in the chain. Separately there is also a map $\nu(n)$ which gives the orientation of the amino acid, and what is hanging off the backbone there.
For some less precise dynamical computations, going through the whole list of all positions can be computationally intensive (a protein can have upwards of tens or hundreds of thousands of amino acids), without being particularly accurate. On the other hand, the basic larger-scale structure of polypeptide chains are fairly well known (classical in the biology literature), and includes things like alpha-helices, beta-pleats, and turns. These three most common structures are all well-approximated by constant torsion and constant curvature space-curves. Therefore, a computationally less demanding way of storing the approximate structure of the protein backbone would be to decompose the folded structure into its "secondary structures", approximate each of those by these space-curves (each can be parametrized completely by the torsion, curvature, total length, starting position, and starting direction). This allows better memory use and faster computations for certain numerical simulations of protein dynamics.
