I have used the following in differential geometry courses. Differential Geometry

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The following notes are on a more elementary level. Elementary Differential Geometry

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The following is a compact treatment of connections and curvature. Connections and Curvature

The following gives a brief discussion of matters related to computing geodesics on a surface in Euclidean space.

Observations on the geodesic flow and its numerical approximation

The following is a (somewhat rough) set of notes on compact Riemann surfaces. The notes were written as a complement to the material on elliptic functions in my Introduction to Complex Analysis and to the material on the Riemann-Roch theorem in Chapter 10 (Dirac Operators and Index Theory) of my PDE text. Notes on Compact Riemann Surfaces

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The following is a brief set of notes on applications of the Riemann-Roch theorem to the index of dbar operators on sections of various holomorphic line bundles and vector bundles over compact Riemann surfaces, with particular attention to the question of when such operators are invertible. Dirac type operators on compact Riemann surfaces

Here is a geometry/PDE approach to the uniformization theorem for Riemann surfaces. Curvature and uniformization (with R. Mazzeo).

Here are some notes on CR manifolds, whose study generalizes the study of hypersurfaces in n-dimensional complex space. CR Manifolds

The second part of the following notes provides a “naive” proof of the index theorem for operators of Dirac type on compact, 2-dimensional manifolds. This includes the Riemann-Roch theorem, which can be regarded as the paradigmatic index theorem.

The following notes relate to index theory for elliptic operators, from the point of view of K-homology. Pseudodifferential operators and K-homology, II

The following are miscellaneous notes on topics related to differential geometry.

- The Hopf bracket (with C. Lebrun)
- Hypoelliptic (and non-hypoelliptic) Hodge theory
- Equivariant isometric embeddings of homogeneous spaces into Hilbert space
- Borsuk’s theorem
- Modular forms and automorphic forms