MATH 782 Dierential Geometry : solutions to homework assignment one
1. Consider a parameterized curve
x = f (v)
v (a, b)
z = g(v)
in the xz-plane, with f (v) > 0 everywhere. Let S be the surface of revolution obtained by
revolving the curve around the z-a
MATH 782 Dierential Geometry : solutions to homework assignment three
1. Calculate the (sectional) curvature of hyperbolic space using the upper half-plane model
U with metric ds2 = (dx2 + dy 2 )/y 2 .
Previously we calculated the Christoel symbols and fo
MATH 782 Dierential Geometry : solutions to homework assignment four
1. Let S be the surface in R3 given by z = y 2 . Calculate the shape operator of S and show
that it is non-zero. [Note: S is isometric to R2 , so both S and R3 are at.]
We can let x and
MATH 782 Dierential Geometry : solutions to homework assignment two
1. Let
be the connection on R2 that has vanishing Christoel symbols k with respect
ij
to Cartesian coordinates (x, y). Calculate the Christoel symbols with respect to polar
coordinates (r
MATH 782 Dierential Geometry : solutions to homework assignment ve
1. A ray is a geodesic : [0, ) M which minimizes the distance from (0) to (t) for
all t [0, ). If M is complete and non-compact, prove that there is a ray starting at
(0) = p for all p M .
MATH 782 Dierential Geometry : solutions to homework assignment six
1. Let M R3 be the helicoid given by
(s, t) (tcoss, tsins, s)
and let S R3 be the surface of revolution given by
(u, v) (vcosu, vsinu, logv).
Show that the map f : M S given in local coor
MATH 782 Dierential Geometry : solutions to homework assignment seven
1. A closed geodesic is an immersion of the circle S 1 in M which satises the geodesic
equation at every point. For example, a great circle on a sphere is a closed geodesic.
Suppose tha