Topics in Differential Geometry. Fall 2015
Overview:
This year (2015) marks the 100th anniversary of Einstein's field equations and the General Theory of Relativity. This would have been impossible without differential geometry and its central notion, the Riemann curvature tensor. In fact the idea of curvature plays an important role in many other areas of mathematics and physics. In this course we shall look at some of its definitions and manifestations in geometry, topology, and mathematical physics. This course has two components: 1.Spectral geometry of Riemannian manifolds, and 2. Curvature invariants in mathematics and physics.
Textbooks:
Topics to be covered:
The following topics will be covered.
- Gauss's theorema egregium. Riemann curvature tensor and its shadows (Ricci and scalar curvature), sectional curvature. Connections on the tangent bundle, Levi- Civita's theorem. Connections on principal G-bundles and on vector bundles. Chern-Weil and Chern-Simons theories. Gauss-Bonnet theorem.
- Spectral geometry and curvature invariants.
- Bi-invariant metrics on Lie groups and Maurer-Cartan equations.
- Einstein-Hilbert action and Einstein's eld equations in general relativity, special solutions, introduction to black holes.
- Gauge theory and Yang-Mills equations, Higgs mechanism, applications to elementary particle physics.
Overview of Spectral Geometry:
Let M be a closed Riemannian manifold and let Δ denote its Laplace operator acting on smooth functions on M. It is a self-adjoint, positive and elliptic differential operator which has a pure point spectrum
0<=λ_1<=λ_2<=... -> infinity
There is also an orthonormal basis \varphi_i , i=1,2,... of L^2 (M) consisting of eigenfunctions of Δ. The spectrum is an isometry invariant of M. Manifolds with the same spectrum, with multiplicities taken into account, are called isospectral. Isometric manifolds are isospectral. Spectral geometry's first goal is to answer the following question:
how much of the geometry and topology of M can be recovered from its spectrum?
For example, is it true that isospectral manifolds are isometric? The essence of this question was captured in the title of a famous article by Marc Kac [8], Can one hear the shape of a drum?
The first giant step in this direction was taken by Hermann Weyl in 1911 [19]. Addressing a conjecture advanced by physicist Lorentz and Sommerfeld around 1910 [9,17], he showed that the dimension and volume of a bounded domain M in R^n, n=2, 3 with smooth enough boundary, is determined by its Dirichlet or Neumann spectrum. In fact he showed that if N(λ)=#{λ_i<= λ} is the eigenvalue counting function, then one has an asymptotic formula
where $\omega_n$ is the volume of the unit ball in R^n. This result, now known as Weyl's Law, and its generalization to closed Riemannian manifolds is often paraphrased as saying:
One can hear the dimension and volume of a Riemannian manifold.
It is now known that one can also hear the total scalar curvature of a closed Riemannian manifold. This kind of results are typically referred to as inverse spectral problems. Results in this direction are both positive and negative.
On the positive side, an infinite sequence of spectral invariants, known as DeWitt--Gilkey--Seeley coefficients can be defined and can be computed, at least in principle, which give information about the geometry of a manifold which solely depends on its spectrum [7]. On the negative side, starting with Milnor's counterexample of 1964 [12], it is known that there are isospectral manifolds which are not isometric. Other milestones in this direction are results by Sunada [18] and Gordon-Web-Wolpert [5]. It was known for sometime that there are non-isomorphic number fields which have the same zeta function. Motivated by this, in [18] Sunada managed to give a general method for constructing isospectral but not isometric manifolds with dimensions bigger than two. In [5], Gordon-Web-Wolpert mangled to construct planar isospectral domains with piecewise linear boundaries which are not isometric.
Time permitting I shall also discuss a sample of direct spectral problems: knowing a manifold geometry, what one can say about its spectrum?
Another set of ideas in spectral geometry concerns with different types of trace formulae and applications to number theory, quantum physics, and quantum chaos. In some sense this even goes back to the very origins of the Weyl's law in quantum mechanics and in deriving Planck's radiation formula [1,9]. Time permitting, this will be touched in some detail in this course.
Topics related to Spectral Geometry:
I am planing to go over some of the following topics:
- Spectral decomposition of L2(M); discreteness of the spectrum, and its finite multiplicity. This needs some Sobolev space theory, functional analysis, and spectral theorem for compact operators.
- Weyl's law for bounded domains in Rn. I shall give a proof very close to Weyl's original proof based on Max-Min principle for eigenvalues of Δ, and the domain monotonicity of eigenvalues.
- Examples: Lattices and at tori, round spheres and spherical harmonics, projective spaces. Milnor's counter example: two lattices in R16 which are isospectral but not isometric. Theta functions and Modular forms for lattices. Gauss circle problem.
- The Minakshisundaram- Pleijel asymptotic expansion for the kernel of the heat operator e^(-tΔ) :
- The scalar functions ai, are invariantly depened on M, are isometry invariants, and their values ai(x) can be explicitly computed in terms of the curvature tensor of M at x and its covariant derivatives. I shall compute the rst three terms. Using a Tauberian theorem of Hardy-Littlewood-Karamata, Weyl's law follows after computing a0(x): a1(x) is 1/6 of the scalar curvature. It follows that the total scalar curvature is determined by spectrum of Δ, i.e.one can hear the total scalar curvature of M. Heat equation proof of Gauss- Bonnet theorem. The formulas for ai(x), for i >3, are exceedingly more and more complicated and it seems at present they are explicitly computed only up to i = 10.
- Spectral zeta function \zeta_Δ(s). Proof of analytic continuation, structure of poles and residues, and special values. Via the Mellin transform, this is more or less equivalent to heat trace asymptotic expansion, but the spectral zeta function has its own merits, and is indispensable, e.g. for determinants of Laplacian and analytic torsion.
- Trace formulae. This is a refinement of Weyl's law and a meeting point for number theory, spectral geometry, and physics in the form of correspondence principles. Poisson summation formula and applications to at tori: one can hear the lengths of closed geodesics of at tori. Semiclassical approximation, Born-Sommerfeld quantization rules and correspondence principles. Selberg trace formula. Relations between trace formulae and Riemann hypothesis. time permiting I shall also look at corresponding results for compact topological groups through examples.
Who shall benefit from this course?
Graduate students and faculty working in Riemannian geometry,
symplectic geometry, geometric analysis and PDE's, mathematical and
theoretical physics, number theory, and noncommutative geometry should
nd this material quite relevant and useful.
Supplimentary Textbooks:
The following texts are standard and I shall use them time to time throughout the course:
References (for spectral geometry):
[1] W. Arendt, R. Nittka, W.
Peter, and F. Steiner;
Weyls Law: Spectral Properties of the Laplacian in Mathematics and
Physics, in Mathematical Analysis of Evolution, Information, and
Complexity. Edited by Wolfgang Arendt and Wolfgang P. Schleich, 2009.
[2] M. Berger, P. Gauduchon, and E. Mazet,
Le
spectre d'une vari'et'e riemannienne. (French) Lecture Notes in
Mathematics, Vol. 194 Springer-Verlag, Berlin-New York 1971.
[3] I. Chavel, Eigenvalues in Riemannian
geometry.
Including a chapter by Burton Randol. With an appendix by Jozef
Dodziuk. Pure and Applied Mathematics, 115. Academic Press, Inc.,
Orlando, FL, 1984.
[4] C. Gordon, and D. Webb, You can't hear
the shape of a drum, American Scientist 84 (JanuaryFebruary): 46-55.
[5] C. Gordon, D. Webb, and S. Wolpert, S.
(1992),
"Isospectral plane domains and surfaces via Riemannian orbifolds",
Inventiones mathematicae 110 (1): 1-22, 1992.
[6] C. Gordon, D. Webb, and S. Wolpert,
"One
Cannot Hear the Shape of a Drum", Bulletin of the American Mathematical
Society 27 (1), 1992.
[7] P. Gilkey, Invariance theory, the heat
equation, and the Atiyah-Singer index theorem.
[8] M. Kac, Can One Hear the Shape of a
Drum? The American Mathematical
Monthly, 73(4), 1-23, 1966.
[9] H. A. Lorentz, Alte und neue Fragen
der Physik. Physikal. Zeitschr., 11, 1234-1257, 1910.
[10] H. P. McKean, Selberg's trace formula
as applied to a compact riemann surface,
[11] H. P. McKean, and I. M. Singer,
Curvature and
the eigenvalues of the Laplacian, J. Differential Geom. Volume 1,
Number 1-2, 43{69, 1967.
[12] J. Milnor, Eigenvalues of the Laplace
Operator on Certain Manifolds. Proc. Nat. Acad. Sci. USA, 51(4),1964.
[13] S. Minakshisundaram, and A. Pleijel,
Some
properties of the eigenfunctions of the Laplace-operator on Riemannian
manifolds. Can. J. Math., 1, 242-256, 1949.
[14] S. Rosenberg, The Laplacian on a
Riemannian Manifold: An Introduction to Analysis on Manifolds (London
Mathematical Society Student Texts).
[15] R. Seeley, Complex powers of an
elliptic operator. Proc. Sympos. Pure Math., Vol. X, pp. 288307, AMS,
1967.
[16] I. M. Singer, Eigenvalues of the
Laplacian and Invariants of Manifolds, Proceedings of ICM, 1974.
[17] A. Sommerfeld, Die Greensche Funktion
der
Schwingungsgleichung furein beliebiges Gebiet. Physikal. Zeitschr., 11,
1057{1066, 1910.
[18] T. Sunada, Riemannian coverings and
isospectral manifolds, Ann. Of Math. 121 (1): 169{186, 1985.
[19] H.Weyl, Uber die asymptotische
Verteilung der
Eigenwerte. Nachrichtender Koniglichen Gesellschaft der Wissenschaften
zu Gottingen. Mathem.-
physikal. Klasse, 110{117, 1911.
[20] H. Weyl, Das asymptotische
Verteilungsgesetz
der Eigenwerte linearer partieller Differentialgleichungen (mit einer
Anwendung auf die Theorie der Hohlraumstrahlung). Mathematische
Annalen, 71(4), 441{479, 1912.
[21] H. Weyl, Uber die Abhngigkeit der
Eigenschwingungen einerMembran von deren Begrenzung. J. Reine Angew.
Math., 141, 1-11, 1912.
[22] H. Weyl, Uber das Spektrum der
Hohlraumstrahlung. J. Reine Angew. Math., 141, 163-181, 1912.
[23] H. Weyl, Uber die Randwertaufgabe der
Strahlungstheorie und asymptotische Spektralgeometrie. J. Reine Angew.
Math., 143, 177-202, 1913.
[24] H. Weyl, Das asymptotische
Verteilungsgesetz der Eigenschwingungen eines beliebig gestalteten
elastischen Krpers. Rend. Circ. Mat. Palermo, 39, 1-50, 1915.
[25] H.Weyl, H. Ramifications, old and
new, of the eigenvalue problem. Bull. Amer. Math. Soc., 56(2), 115-139,
1950.
[26] E. Witt, Eine Identitat zwischen
Modulformen zweiten Grades. Abh. Math. Sem. Hansischen Universitat, 14,
323-337, 1941.