Vorlesung: Differential Geometry II/Differentialgeometrie II (summer semester 2021)
Moodle (key: riemann),
AGNES.
The lectures takes place on Wednesday and Thursday 11:15–12:45.
The tutorial takes place on Thursday 13:15–14:45.
The Zoom invite can be found on the Moodle page.
Prerequisites
This course assumes some familiarity with Differential Geometry.
If you have taken Differential Geometry I in WS20/21, then you are more then well-prepared.
Lecture Notes
Here are my live lecture notes.
Topics
This is a course on Riemannian Geometry.
I plan to cover the following topics (not necessarily in this order):
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What is a Riemannian metric?
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Covariant derivatives,
the Levi-Civita connection,
the Fundamental Theorem of Riemannian Geometry
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Geodesics, geodesic flow, the exponential map
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Riemannian curvature; sectional, Ricci, scalar curvature
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Second variation and
Jacobi fields
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Spaces of constant curvature,
the Riemann–Killing–Hopf Theorem
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Notions of completeness,
Hopf–Rinow Theorem
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Spaces of non-positive curvature,
Hadamard's Theorem
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A first glimpse at the interaction between topology and geometry:
Gauß–Bonnet theorem
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Bonnet–Myers Theorem
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The Bochner technique:
Killing fields;
harmonic forms;
Hurwitz' automorphism theorem for Riemann surfaces
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The Lichnerowicz–Obata Theorem
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Comparison geometry;
Rauch's Theorem, Bishop–Gromov, applications
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Holonomy
Depending on the students' interest (and time permitting) I might also discuss some of the following:
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Morse theory of energy functional
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Chern–Weil theory and
the Chern–Gauß–Bonnet theorem
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Cheeger–Gromoll splitting theorem
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Hodge theory
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Gromov's Betti number bounds
If you have any questions or suggestions, then feel free to email me at walpuski@math.hu-berlin.de