Differentiable manifolds

Tue. 12:30-14:30 Amado 719, Wed. 16:30-17:30 Amado 919 

    • The theory of smooth manifolds deals with objects which locally look like Euclidean spaces (e.g. smooth curves and surfaces) but may have a complicated global structure. It plays an important role in many areas of modern mathematics and physics, since most notions initially defined for Euclidean spaces can be easily generalized to their look-alikes. In particular, almost all notions and theorems of multivariable calculus generalize to smooth manifolds. This course is an introduction to smooth manifolds, their maps, and differential and integral calculus on manifolds.

Prerequisites: 

      • Linear algebra (bases, matrices, linear operators, change of bases, ranks of linear maps, quadratic forms)
      • Multivariable calculus (functions of several variables, directional derivatives, multiple integrals, change of variables and Jacobians, implicit and inverse function theorems)
      • Basic notions of topology (open/closed sets, compactness, Hausdorff spaces, continuous maps and homeomorphisms).

Syllabus:

      • Definitions of smooth manifolds; maps, atlases. Examples: projective spaces, grassmanians, etc. Smooth structures. Smooth maps between manifolds. Manifolds with boundary.
      • Tangent space and tungent bundle. Differential of a smooth map. Immersions and embeddings. Submanifolds. Implicit function theorem.
      • Technicalities: Partition of unity, approximations of continuous maps by smooth and analytic ones, transversality, Morse-Sard theorem, strong transversality theorem.
      • Whitney theorem on immersions and embeddings of compact manifolds into Euclidean space.
      • Orientation of a manifold. Degree of a map between compact smooth oriented manifolds. Sphere eversion.
      • Smooth tensors on manifolds. Vector fields, differential forms, Riemannian metrics. Existence of Riemannian metrics.
      • Critical points. Euler characteristics. Basics of the Morse theory. Gluing handles.
      • Exterior product of differential forms, differential of differential forms, closed and exact forms, Poincare lemma on local exactness of any closed form, De Rham cohomology.
      • Integration of differential forms over smooth manifolds, Stokes theorem.

Final exam and grades: 

    • There will be several (3-4) homework assignments, which will sum up to approx. 50% of the final grade. The final exam will consist of a hand-out assignment given for few weeks; the exam will add the remaining 50% of the ?final grade.

Recommended Books: 

  • I will use different books for different topics. In particular, V. Guillemin, A. Pollack, Differential topology (Prentice-Hall, 1974) is a good one and covers a large part of the course.Some others which I’ll use are G. Bredon, Topology and geometry (Springer-Verlag, 1993), W. M. Boothby, An introduction to differentiable manifolds and Riemannian geometry (second.ed. Academic Press 1986), B. Dubrovin, A. Fomenko, S. Novikov, Modern geometry (Springer, 1992), J. Lee, Introduction to smooth manifolds (Springer-Verlag 2003), A. Wallace, Differential topology; first steps (Benjamin, 1968).

Homework Assignments: