The differential geometry qualifying exam covers the calculus of differentiable manifolds and the basic structure of Lie groups, the geometry of curves and surfaces in Euclidean space, and the differential geometry of smooth manifolds. Specifically:

Calculus of differentiable manifolds:

1. Manifolds, tangent vectors, differential forms and vector fields. Tensors. [Warner, 1, 2; Spivak I, 4; Boothby, 1, 3-5]
2. Submanifolds and the implicit function theorem, local coordinate systems. [Warner, 1; Spivak II, 2; Boothby, 2]
3. The Frobenius Theorem, foliations and distributions. [Warner, 1; Spivak I, 6; Boothby, 1, 5]
4. The calculus of differential forms, including Stokes' theorem, the Lie derivative, the Poincare lemma, de Rham cohomology, integration over forms. [Warner, 2, 4; Spivak I, 7-8; Boothby, 6]
5. Lie groups and Lie algebras, subgroups, homomorphisms, the exponential map, continuous homomorphism theorem and closed subgroup theorem, the adjoint representation, homogeneous spaces. [Warner, 3; Spivak I, 10; Boothby, 4]
6. Basic theory of vector and fiber bundles. [Warner, 1,2; Johnson, 6]

1. Curves in space, including curvature and torsion, the Frenet formulas, the isoperimetric inequality, the four-vertex theorem, Fenchel's theorem. [Johnson, 1; do Carmo, 1; Spivak II, 1]
2. Surface theory, including principal curvatures, the Gauss and mean curvatures, geodesics on surfaces and parallelism, conjugate and cut loci. [Johnson, 2; do Carmo, 2-4; Boothby, 8]
3. The Gauss-Bonnet theorem, both in local form and globally. [Johnson, 4; do Carmo, 4; Spivak III, 6]

1. Riemannian metrics, parallelism, Levi-Civita connections and geodesics on manifolds. [Johnson, 3; Boothby, 7; Spivak III, 6]
2. The Riemann curvature tensor, second fundamental form of submanifolds, structure equations, and the algebra of curvature operators. [Johnson 4, 5; Boothby, 7; Spivak II, 5]
3. Connections on principal bundles, Bianchi identities and relationship to Riemannian geometry. [Johnson,. 6; Spivak II, 8]

References to specific authors above refer to the following texts. The lecture notes by Johnson are available from the Mathematics Department.

1. W. Boothby. An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press, 1975.
2. M. Do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall. 1976.
3. D.L. Johnson, Differential Geometry, Math 424 Course Notes, Lehigh Mathematics Department. Xerox Machine, 1996.
4. Mike Spivak, A Comprehensive Introduction to Differential Geometry, vol I-V, Publish or Perish. 1970-1975.
5. F. Warner, Foundations of Differential Manifolds and Lie Groups, Springer-Verlag, 1980.

1. N. Hicks, Notes on Differential Geometry, Van Nostrand, 1965.
2. S. Kobayashi and K. Nomizu, The Foundations of Differential Geometry, vo l I-II, Wiley. 1963.
3. F. Morgan, Riemannian Geometry. a Beginner's Guide, Jones & Bartlet, or A.K. Peters, 1997.