Differential Geometry II - Marc Burger - 2016
Please sign with your name and the date on which you had your exam. If you use this wiki, contribute to it as well or terrible things will happen to you: like waking up on Sunday and realizing that it is actually Monday.
24.08.2016 13.00 - 13.30 Wendel
B: Give the definition of a parallelizable manifold.
W: (definition)
B: Is the n-torus parallelizable?
W: Yes: Using the equivalence of parallelizability and the existence of a collection of smooth vector fields that at each point p form a basis of \( T_pM \). We identify \( T^n = R^n/Z^n \) and then the covering map gives us such a family. (I struggle how exactly such a covering map gives the desired family)
B: How does a smooth vector field on \( T^n \) look like?
W:(I don't understand what he means by look like)
B: Can you mention an example of such a vector field?
W: Any vector field? The zero vector field.
B: That doesn't help the argument. Draw a non-trivial example for n = 2.
W: A vector field on \( R^n \) must be periodic such that the push-forward of that vector field along the covering map is well-defined. (Struggle to draw a vector field on the square such that tangent vectors on identified borders are equal)
B: No just think of constant non-zero vector fields. Give the definition of a Riemannian manifold.
W:(definition)
B: Give the definition of a geodesic.
W:(definition)
B: What is \( \Gamma(c^*TM) \)?
W: pull-back of tangent bundle along c
B: What is \( D/dt \)?
W:(properties defining \( D/dt \) )
B: The product rule does not work this way.
W: (corrects product rule)
B: \( f'*X \) should be \( X(f) \). No you are right.
W: \( X(f) \) is the case of the generalization to parametrized manifolds.
B: Characterize geodesics using length/distance.
W: (geodesics are locally length minimizing curves).
B: What kind of curves are we considering?
W: piece-wise \( C^1 \)
B: What about globally length minimizing curves.
W: Globally minimizing implies by locally minimizing by (sketch of proof) implies geodesic.
B: There is another characterization of geodesics as critical points of a functional.
W: (def of energy and variation (with fixed endpoints), 2nd part of the First variation formula theorem ).
B: How would you check this condition.
W: The same theorem gives a formula for \( dE(c_t)/dt \) at 0.
B: State the formula.
W: I don' t know the formula.
B: What does Bonnet-Myers say?
W: If sectional curvature is ...
B: That is Synge. Bonnet-Myers is the one about Ricci curvature of connected mfd.
W: (statement of Bonnet-Myers)
B: Now there is still an assumption missing.
W:(I don't remember what we already stated and I don't remember the theorem very well.)
B: It must be complete to get the proof started.
W: Ah yes: Any of the four equivalent properties in Rinow-Hopf and then it suffices to consider only minimal geodesics.
(alarm clock rings)
B: We are done. Do you have any other exams?
W: I have an exam on graph theory.
B: Graph theory is beautiful.
25.08.2016 09:00-09:30 Nicola
B: Definition of parallelizable manifold, show that \(S^3\) is parallelizable
N: It is, as a Lie group (unit quaternions), then I show that any Lie group is parallelizable
B: What is \( \Gamma(c^*TM) \)? What is \( D/dt \)? Give the definition of a geodesic. How can we characterize geodesics with a functional?
N: Local minima of energy functional; I define variations,... but can't remember exactly the first variation formula, so he helps me to recover it.
B: Is the universal cover of a compact Lie group always compact?
N: No, e.g. \(T^2\) has universal cover \(\mathbb{R}^2\).
B: Are there conditions to ensure this?
N: Yes, as a corollary of Bonnet-Myers we have that... (Corollary 3.21)
B: Can you prove this?
N: Yes (with some help for the details)
