M462 (HANDOUT 5)
0.1. Gauss Map.
Denition 0.2. A regular surface S is called orientable if it possesses a unit normal
vector eld N (p). Such a surface possesses precisely two unit normal elds given
by N (p). A choice of one of then is called an orientatio
M462 (HANDOUT 5)
0.1. The rst fundamental form. Recall that an inner product on Rn , or more
generally a nite dimensional vector space, is an operation , : Rn Rn R such
that
u, v = v , u
u, a1 v1 + a2 v2 = a1 u, v1 + a2 u, v2
if u = 0, u, u > 0
Any inner
M462 (HANDOUT 4)
0.1. Parameterized surfaces (continued).
Example 0.2. Given a curve x = f (v ), z = g (v ), a < v < b in the xz -plane, we get
a surface by revolving this around the z -axis. This is parameterized by F (u, v ) =
(f (v ) cos u, f (v ) sin
M462 (HANDOUT 3)
0.1. Surfaces. Recall that subset U Rn is open if all points in U are interior
points. This means that if p U , then we can a nd a (small) > 0 such that all
points in Rn of distance less than from p lie in U . For example the interior of
M462 (HANDOUT 2)
0.1. Vector product. Given vectors U, V, W R3 , recall that they form a basis
if and only det(U, V, W ) = 0. If the determinant is positive, then the ordered basis
U, V, W is called right handed, otherwise it is called left handed. Right
M462 (HANDOUT 1)
0.1. Essential Information. .
Professor: Donu Arapura
Oce: Math 642
Oce hrs: TBA
Book: do Carmo, Dierential Geometry of Curves and Surfaces
There may be some additional notes on the web at www.math.purdue.edu/dvb
Grading Policy: Grade bas
Introduction to dierential forms
Donu Arapura
May 27, 2012
The calculus of dierential forms give an alternative to vector calculus which
is ultimately simpler and more exible. Unfortunately it is rarely encountered
at the undergraduate level. However, the
RIEMANN SURFACE PROBLEM LIST
There are close to 40 problems of various types. Most are elementary, but some
are not. Take a look at the list, and nd something that appeals to you and that you
think you can do. Eventually Im going to ask you to present eit
M462 (HANDOUT 9)
0.1. Christoel symbols. Let S be a regular parametrized surface. Set
x y z
Su = ( ,
,
)
u u u
x y z
,
)
Sv = ( ,
v v v
(We called these and previously.) We can form the second derivatives
2x 2y 2z
,
,
)
u2 u2 u2
etc. These expressions are
M462 (HANDOUT 8)
0.1. Gaussian curvature. Recall that when S is regular parametric surface, with
parameters u, v , we have
Nu = a11 + a12
Nv = a21 + a22
where N is the unit normal, and = (x/u, . . .), . . . We can read o the Gaussian
curvature as det(ai
M462 (HANDOUT 7)
0.1. Gauss Map (continued). Let S be a regular surface given parametrically
by x = f1 (u, v ), y = f2 (u, v ), z = f3 (u, v ). Let p = (f1 (0, 0), . . .). Our goal is
to calculate the various curvatures and the second fundamental form |p