A geometric classification of the compact four-dimensional Einstein-Weyl manifolds with at least four-dimensional symmetry group is given. Our results also sharpen previous results on four-dimensional Einstein metrics and correct Parker's topological classification of cohomogeneity-one four-manifolds.

Let M be a compact Criemannian manifold of nonpositive curvature1 and with fundamental group . It is well known [8, p. 102] that M is a K (, 1) and thus completely determined up to homotopy type by . In light of this fact it is natural to ask to what extent the riemannian structure of M is determined by the structure of , and the intent of this paper is to demonstrate that rather strong implications of this sort exist. In the case that M has strictly negative curvature, the group is known to be highly noncommutative. Every abelian, in fact, every solvable, subgroup of is cyclic [3]. It is therefore a plausible conjecture that in the nonpositive curvature case, will possess large amounts of commutativity only under special geometric circumstances. We shall show that this is true, that indeed those properties of which involve commutativity have a dramatic reflection in the riemannian structure of M.

Let M be a complete two-dimensional surface immersed into the three-dimensional Euclidean space. Then a classical theorem of Hilbert says that when the curvature of M is a non-zero constant, M must be the sphere. On the other hand, when the curvature of M is zero, a theorem of Hartman-Nirenberg [4] says that M must be a plane or a cylinder. These two theorems complete the classification of complete surfaces with constant curvature in R 3.

THE MAIN PURPOSE of this paper is to give a simple and unified new proof of the Witten rigidity theorems, which were conjectured by Witten and first proved by Taubes [25], Bott-Taubes [S], Hirzebruch Cl23 and Krichever [15]. Our proof shows that the modular invariance, which is the intrinsic symmetry of elliptic genera, actually implies their rigidity. Some new properties of elliptic genera and their relationships with theta-functions are also discussed. We remark that our proof makes essential uses of the new feature of loop groups and loop spaces, the modular invariance. We note that, with the help of the modular group, we can catch the topological information on loop space by simply working on finite-dimensional manifold. By developing this idea further, in [21] we have proved the rigidity of the Dirac operator on loop space twisted by higher-level loop group representations, while the Witten ridigity theorems are the special cases of level 1. Many topological vanishing theorems are also derived in [21] by refining the argument in this paper, especially an &-vanishing theorem for loop space. In [19] modular invariance is used again to establish a general miraculous cancellation formula, relating the Hirzebruch L-form to certain twisted &-forms, which has many interesting topological results as consequences. These results were announced in [20].

All orientable metric surfaces are Riemann surfaces and admit global conformal parameterizations. Riemann surface structure is a fundamental structure and governs many natural physical phenomena, such as heat diffusion and electro-magnetic fields on the surface. A good parameterization is crucial for simulation and visualization. This paper provides an explicit method for finding optimal global conformal parameterizations of arbitrary surfaces. It relies on certain holomorphic differential forms and conformal mappings from differential geometry and Riemann surface theories. Algorithms are developed to modify topology, locate zero points, and determine cohomology types of differential forms. The implementation is based on a finite dimensional optimization method. The optimal parameterization is intrinsic to the geometry, preserves angular structure, and can play an important role in various applications