Spring 2010, MWF 1-2, Science Center 411
My name: Andrew Cotton-Clay (please call me Andy)
Office: Science Center 527
Office Hours: Tu 4:30-5:30 and W 2-3 (tentative).
E-mail: acotton at math
Syllabus: Math 230b Syllabus
Course iSite: Math 230b iSite
A continuation of Mathematics 230a. Topics in differential geometry: Analysis on manifolds. Laplacians. Hodge theory. Spin structures. Clifford algebras. Dirac operators. Index theorems. Applications.
Note: References are as in the syllabus.
Mon 1/24: Review of Riemannian manifolds, gradient, Levi-Civita connection. Hodge star operator and
codifferential (see also Math 230a, problem 12.2). Hodge laplacian.
Divergence of a vector field as trace of the full covariant derivative. On functions, Hodge laplacian is
minus the divergence of the gradient (modulo lemma for next time that (div X) dvol = L_X dvol).
[P, 2.1, 7.2, 7.3.1].
Wed 1/26: Riemannian volume form in coordinates. Proof that (div X) dvol = L_X dvol. Computation of
Laplacian in coordinates, plus example of flat metric in polar coordinates. Sketch of the analysis
required to prove the Hodge decomposition of k-forms into ker(Laplacian) and im(Laplacian). Proof
from this that the space of harmonic k-forms is isomorphic to the k-th de Rham cohomology.
[P, 2.1, 7.2]. See also [LM, III.5, Theorem 5.5] for the analysis (we'll cover this later).
Fri 1/28: Poincare duality from the Hodge isomorphism. Bochner formula for harmonic functions in Euclidean
space, for harmonic functions on Riemannian manifolds, and for harmonic 1-forms on Riemannian manifolds.
Corollary that Ric >= 0 implies harmonic 1-forms are parallel, and implies that b_1(M) <= dim(M), with equality
if and only if M,g is a flat torus.
[P, 7.3.1].
Mon 1/31: Killing vector fields: definition, generate isometries, determined by value and covariant derivative
at a point. Full covariant derivative of such is skew-symmetric as (0,2)-tensor. Bochner formula for Killing
vector fields. Corollary that Ric <= 0 implies Killing vector fields are parallel, and Ric < 0 implies there are
no nontrivial ones. Second covariant derivative on vector bundles over Riemannian manifolds. Hessian of a
function as an example. Definition of connection Laplacian as minus the trace of the second covariant derivative.
[P, 7.1, 7.3.2]. See also [LM, II.8] for second covariant derivative and connection laplacian.
Wed 2/2: Connection laplacian agrees with \nabla^* \nabla, and thus is non-negative, symmetric. Reformulation of
Bochner method in terms of connection laplacian. Formula for d in terms of covariant derivatives, and formula for
\delta in normal coordinates at a point. Proof of Bochner formula for harmonic 1-forms in terms of connection
Laplacian. [P, 7.3.2].
Fri 2/4: Complex manifolds, decomposition of complex tangent space, (p,q)-forms. Hodge star for a Hermitian metric.
Definition of Dolbeault cohomology and Hodge isomorphism for del and dbar Laplacians. Example computation of dbar
Laplacian on C^n with Euclidean metric acting on functions. (For the lectures on complex and Kahler manifolds, see
e.g. Griffiths and Harris, Principles of Algebraic Geometry, Chapter 0, or Huybrechts, Complex Geometry,
Chapters 1 and 3.)
Mon 2/6: Linear algebra of Hermitian vector spaces: Isomorphism of real vector space with (1,0) vector space.
Definition of omega from metric on real vector space for which J is an isometry. The form $g-i\omega$ is Hermitian
and on the real vector space is twice the g_{\mathbb{C}} on the (1,0) vector space under the isomorphism. Definition of L and Lambda,
sl_2 relations, and computation of [L^k,\Lambda].
Wed 2/8: Lefschetz decomposition for a Hermitian vector space. Proof that the d-Laplacian, the del-Laplacian,
and the dbar-Laplacian agree up to a factor of 2 for Kahler manifolds (assuming formula in dollar
signs in Fri 2/10 summary). Hodge and Lefschetz decompositions for Kahler manifolds. Symmetries of the Hodge diamond.
Computation of the (1,1)-form omega in coordinates.
Fri 2/10: An Hermitian metric on a complex manifold is Kahler if and only if there are complex coordinates in which it
osculates to second order with the Euclidean metric. Computation for C^n with Euclidean metric showing that
$[\Lambda,\partial] = i\overline\partial^*$, with corollary that this holds in Kahler manifolds. S^3 x S^1 as complex
but not Kahler manifold. H^(p,p) and thus H^2p is nonzero for Kahler manifolds.
Mon 2/14: Kahler potentials, example of flat C^n. Fubini-Study metric on CP^n. Differential operators on a
manifold X between vector bundles E and F. The principal symbol of an order m differential operator as a section of
Sym^m TX tensor Hom(E,F). Identification of Sym^m TX_p with degree m homogeneous polynomials on T^*X_p. Definition
of elliptic, and the Hodge Laplacian as an example. [LM, 3.1]
Wed 2/16: Sobolev spaces L^2_k, k nonnegative integer, via completion using L^2 norms of first k derivatives. Fourier transform on R^n,
Schwarz space of rapidly decreasing functions. Properties of Fourier transform, inversion formula, Plancherel
formula. Definition of L^2_s, s real, via Fourier transform. C^k norm and Sobolev embedding theorem. Rellich lemma
for Sobolev spaces. [LM, 3.2]
Fri 2/18: Duality of L^2_s and L^2_{-s}. Sobolev spaces L^2_s(E) for a vector bundle E over a manifold and properties.
Pseudodifferential operators of order m: definition, give bounded linear operators L^2_s to L^2_{s-m}. [LM, 3.2-3]
Wed 2/23: Formal developments of pseudodifferential operators (sums convergent modulo smoothing).
Existence/uniqueness up to infinitely smoothing operators given a formal development. Formal developments
for certain integral operators. Corollary that compactly supported pseudodifferential operators are
local modulo smoothing operators. [LM, 3.3]
Fri 2/25: Formal developments for adjoints and compositions of compactly supported pseudodifferential
operators. Elliptic pseudodifferential operators are invertible modulo smoothing
operators. Solutions to smooth elliptic equations are smooth. [LM, 3.3-4]
Mon 2/28: Fredholm operators and index: definition, examples remarks on properties. Operators invertible modulo
compact are Fredholm. Elliptic pseudodifferential operators on compact manifolds are Fredholm, with well-defined
index independent of Sobolev space. Hodge decomposition for self-adjoint elliptic operators on compact manifolds.
[LM, 3.5]
Wed 3/2: Eigenfunctions of self-adjoint elliptic operators are smooth, span L^2, have discrete eigenvalues with
at most polynomially many (with multiplicity) up to a given value. Properties for a heat kernel. Expression
in terms of eigenfunctions and uniform C^k convergence thereof. Trace of the heat operator. Index of an elliptic
operator P in terms of the traces of the heat operators for P*P and PP*. [LM, 3.5-6], [BGV, 2.6]
Fri 3/4: Laplacian in exponential coordinates in terms of the determinant of dexp, radial derivatives, and the
Laplacian on spherical shells. Laplacian of a product. Overview of heat kernel on a Riemannian manifold and
Weyl's theorem. [BGV, 2.1 (and overview of 2.2-6)]
Mon 3/7: Heat kernel on R^n and check of properties. Approximate heat kernel on a Riemannian manifold and formal
expansion as t approaches zero. Sketch of formally recovering the full heat kernel therefrom. Leading term in t
of sum over eigenvalues of exp of -t times eigenvalue. [BGV, 2.2-5]