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Submersions, imersions, submanifolds, Whitney Theorem. Lie Theory: Vector fields, Lie brackets, Lie derivative.

Differential Geometry: Local Curve Theory

Distributions and Frobenius Theorem. Lie groups, Lie algebras, actions. Differential Forms: Tensor and exterior algebras, differential forms. Orientation, integration over manifolds, homotopy. Stokes Theorem, Mayer-Vietoris sequence. Fiber Bundles: Vector bundles, connections, curvature, metrics.

Parallel transport, Riemannian manifolds, geodesics. Characteristic classes, Chern-Weil theory. Gauss-Bonnet Theorem.

Publication Information

Gray, A. Kreyszig, E. Lipschutz, M. Theory and Problems of Differential Geometry. New York: McGraw-Hill, Spivak, M.

Mathematical Sciences Research Institute

Struik, D. Lectures on Classical Differential Geometry.

  1. Differential Geometry and its Application?
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  5. [] Differential Geometry of Curves and Surfaces in Lorentz-Minkowski space?

Weatherburn, C. Differential Geometry of Three Dimensions, 2 vols. Our far-reaching generalization of the classical work of Delaunay classified all complete constant mean curvature surfaces admitting a one-parameter group of isometries; the new infinite families of such surfaces generated by this work are currently of interest in other areas of surface theory. Classical surface theory is the study of isometric immersions of surfaces into Euclidean 3-space. In this study the umbilic points have a special significance both topologically and geometrically and the Caratheodory conjecture of eighty years standing is one of the most resistant of problems in this area.

Beginning with a generic geometric solution to this conjecture and the establishing of a remarkable connection with the theory of compressible plane fluid flow, we have made profound contributions to our understanding of this phenomenon, so that these purely mathematical results are now being applied to the solution of fundamental problems in the theory of relativity. Our current research on this area focuses on complex manifolds with non-positive curvature, exhibiting various manifestations of hyperbolicity and parabolicity.

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Much of the progress in Riemannian geometry that took place over the last decades has been made via the use of deep analytic techniques on non-compact manifolds. The central object of study is the Laplace operator, acting on functions and on differential forms.

Outline of Course

Our work on the spectral theory of the Laplacian uses techniques from quantum mechanical scattering theory. A recent example has been one proof that the Laplacian of the 4-dimensional hyperbolic space is rigid, in the Hilbert space sense. Probabilistic methods, coming from the theory of Brownian motion, have also been used with success in our discovery of a new family of Liouville manifolds having a positive lower bound for the Laplacian spectrum; these manifolds provided counter-examples to a conjecture of Schoen and Yau on Liouville manifolds.

Over the last thirty years Gromov has made important contributions to diverse areas of mathematics and pioneered new directions in mathematics such as filling Riemannian geometry, almost flat manifolds, word-hyperbolic groups, Carnot geometry and applications to the rigidity of symmetric spaces, to name but a few. In the past ten years it has been observed that there are profound connections between the existence of metrics with positive scalar curvature on a given compact space and the topological structure of the space.

An outstanding problem in this area is the existence of metrics of positive scalar curvature on compact spin manifolds.

The expert in this area at Notre Dame successfully solved this important problem by a detailed study of positive scalar curvature metrics on quaternionic fibrations over compact manifolds.