Exercise 4.1 — Harmonic functions (10 pts). Let M be a compact and connected Rie
mannian manifold. A function u: M ! R (i.e. a 0-form u 2 Ω0„M”) is said to be har
−∆u = δdu = 0: (1)
(a) Show that if @M = then every harmonic function is constant. (You can use the
fact that u is constant if and only if du = 0 if and only if kduk2 = 0.)
(b) Give an example of a non-constant harmonic function when M has boundary. (You
can pick your own M.)
Hint ⟪dα; β⟫ = ⟪α; δβ⟫ if M has no boundary.
The Foucault pendulums are large and heavy pendulums displayed in several science museums
over the world. They swing like a usual pendulum in a vertical plane. After several hours (say
a day), this vertical plane will turn with a certain angle. This is the result of the Gauss–Bonnet
theorem for parallel transported tangent vector.
The direction of the pendulum is a tangent vector of the earth. The rotation of the globe
brings the entire device (the museum building) along the circle of constant latitude counted
from the equator), and in the least dissipative manner (adiabatic process) the tangent vector
(pendulum’s swinging direction) is parallel transported along this circle.
Exercise 4.2 — Foucault pendulum (10 pts). UCSD is locatedapproximatelyat32:9◦ N,
117:2◦ W. If we set up a Foucault pendulum in UCSD, in how many degrees (clockwise or
counterclockwise?) would the swing direction rotate after a full rotation of the earth (i.e. in
23 hours 56 minutes 4.091 seconds)?
Connections on Vector Bundles
We have a descent idea of what a tangent vector eld on a manifold is. A tangent vector eld
X 2 Γ„TM” on a manifold M is an assignment of a tangent vector Xp 2 TpM at each point
p 2 M. Since at dierent points p;q 2 M the vector spaces TpM;TqM are two dierent spaces,
a priori there isn’t a comparison like “Xq − Xp” we can make to measure derivative of a tangent
vector eld. That’s why we have covariant derivative, also called connection derivative, for
measuring rate of change of a tangent vector eld. On a manifold with metric (Riemannian
manifold), there is a canonical Levi-Civita connection for this connection. The discovery of
the uniqueness of the Levi-Civita connection (fundamental theorem of Riemannian geometry)
basically removes all the ambiguity of what the rate of change of a tangent vector means.
What about a more general vector-valued eld that is not necessarily tangent vector eld?
In the general case there is not a unique connection like Levi-Civita connection.
A vector bundle E over M is the union of vector spaces E = —p2M Ep based at p. (Of
course, an example is Ep = TpM and E = TM.) A section 2 Γ„E” is an assignment of
p 2 Ep per point p 2 M. (For example, a tangent vector eld is a section of the tangent bundle
E = TM.)
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