MATH661 DIFFERENTIAL GEOMETRY II
SYLLABUS
RENATO GHINI BETTIOL
Meeting coordinates: Tuesdays and Thursdays, 3pm-4:30pm at DRL 4C4
This course will focus on two central areas of modern Differential Geometry:
(1) Lie groups and isometric actions. We will develop the basic theory of
compact Lie groups and their actions on manifolds. Topics to be covered
include the fundamental theorems relating Lie groups and their Lie alge-
bras, bi-invariant Riemannian metrics, Killing form, maximal tori, roots,
Weyl group, Dynkin diagrams, isometric and proper actions, slice theo-
rem, principal and associated bundles, stratification by orbit types, and
homogeneous spaces. The basic reference for this material is [1, Ch. 1-4].
Individual copies of these chapters will be provided for everyone.
(2) Comparison geometry. The central results to be discussed are the compari-
son theorems of Rauch and Toponogov for sectional curvature, and Bishop-
Gromov and Heintze-Karcher for Ricci curvature, together with some appli-
cations and related results, including the Grove-Shiohama diameter sphere
theorem, Gromoll-Meyer soul theorem, and Gromov’s theorem on Betti
numbers. The main reference for these topics is [2], besides the original
research articles containing the aforementioned results.
(3) Extra topics (time permitting). According to the interests of the audience,
if time permits, we might also discuss topics including: cohomogeneity one
manifolds, manifolds with positive curvature, general Weitzenb¨ock formulas
and the Bochner technique, spectral geometry of the Laplacian, etc.
There will be no assignments/exams for this course; instead, the plan is to point
out open research problems connected to the material and discuss recent progress.
References
[1] M. M. Alexandrino and R. G. Bettiol,Lie Groups and Geometric Aspects of Isometric
Actions. Springer, Cham, 2015. ISBN: 978-3-319-16612-4; 978-3-319-16613-1
[2] J. Cheeger and D. G. Ebin,Comparison theorems in Riemannian geometry. Revised reprint
of the 1975 original. AMS Chelsea Publishing, Providence, RI, 2008. ISBN: 978-0-8218-4417-5
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