Math 5378, Differential Geometry
Practice questions for Test 2
The exam itself will be closed book, no notes.
Note: There are more practice questions appearing here than would appear
on an actual exam. The actual exam will have five questions, and two of
them will be off this list.
Solutions will be posted on Monday, April 28.
1. Find all possible tra jectories of the vector field w(x, y) = (−y, x) on
R2.
2. If the first fundamental form in coordinates is given by E=eu, F =
0, G =ev, find a vector field of unit length perpendicular to the vector
field xu−xv.
3. If f:S1→S2is an isometry between surfaces and α(s) : (a, b)→S1
is a geodesic parametrized by arc length, show that f(α(s)) is also a
geodesic parametrized by arc length.
4. Suppose xis a coordinate chart on a surface, with coefficients E, F,
and Gof the first fundamental form. Prove the following identities.
hxuu, xui=1
2Eu
hxuu, xvi=Fu−1
2Ev
Use these to show the matrix identity
1
2Eu
Fu−1
2Ev=E F
F GΓ1
11
Γ2
11
5. Prove that the sphere of radius R > 0 centered at the origin has con-
stant Gaussian curvature 1/R2and mean curvature −1/R.
6. Suppose (u(s), v(s)) is a curve in R2and xis a coordinate chart so
that x(u(s), v(s)) is a curve parametrized by arc length. Write down
the conditions on uand vnecessary for this curve to be a geodesic in
the surface.
