Joel L. Weiner

Affine differential geometry of surfaces in ℝ4

Employing the method of moving frames, i.e. Cartans algorithm, we find a complete set of invariants for nondegenerate oriented surfacesM2 in ℝ4 relative to the action of the general affine group on ℝ4. The invariants found include a normal bundle, a quadratic form onM2 with values in the normal bundle, a symmetric connection onM2 and a connection on the normal bundle. Integrability conditions for these invariants are also determined. Geometric interpretations are given for the successive reductions to the bundle of affine frames overM2, obtained by using the method of moving frames, that lead to the aforementioned invariants. As applications of these results we study a class of surfaces known as harmonic surfaces, finding for them a complete set of invariants and their integrability conditions. Further applications involve the study of homogeneous surfaces; these are surfaces which are fixed by a group of affine transformations that act transitively on the surface. All homogeneous harmonic surfaces are determined.

A rigidity theorem for the Clifford tori in

Let S3 be the unit hypersphere in the 4-dimensional Euclidean space R4 defined by ∑4 i=1 xi = 1. For each θ with 0 < θ < π/2, we denote by Mθ the Clifford torus in S 3 given by the equations x1 + x 2 2 = cos 2 θ and x3+x 2 4 = sin 2 θ. The Clifford torus Mθ is a flat Riemannian manifold equipped with the metric induced by the inclusion map iθ : Mθ → S3. In this note we prove the following rigidity theorem: If f : Mθ → S3 is an isometric embedding, then there exists an isometry A of S3 such that f = A ◦ iθ. We also show no flat torus with the intrinsic diameter ≤ π is embeddable in S3 except for a Clifford torus.

The Principle of Stationary Action in Biophysics: Stability in Protein Folding

We conceptualize protein folding as motion in a large dimensional dihedral angle space. We use Lagrangian mechanics and introduce an unspecified Lagrangian to study the motion. The fact that we have reliable folding leads us to conjecture the totality of paths forms caustics that can be recognized by the vanishing of the second variation of the action. There are two types of folding processes: stable against modest perturbations and unstable. We also conjecture that natural selection has picked out stable folds. More importantly, the presence of caustics leads naturally to the application of ideas from catastrophe theory and allows us to consider the question of stability for the folding process from that perspective. Powerful stability theorems from mathematics are then applicable to impose more order on the totality of motions. This leads to an immediate explanation for both the insensitivity of folding to solution perturbations and the fact that folding occurs using very little free energy. The theory of folding, based on the above conjectures, can also be used to explain the behavior of energy landscapes, the speed of folding similar to transition state theory, and the fact that random proteins do not fold.