CHAPTER 4
Geometry of lengths and distances
1
Lecture 36
Lets start by looking at standard Rn . Straight lines are distinguished by
being the shortest lines joining two points. More precisely,
Lemma 36.1. Let : [a, b] Rn be a smooth path, with (a) = p and
CHAPTER 1
Local and global geometry of plane curves
1
Lecture 1
Terminology from linear algebra: the scalar product of X, Y R2 is
X, Y = X1 Y1 + X2 Y2 .
The length of a vector is
X = X, X 1/2 .
The rotation by any angle is the linear transformation of R2
CHAPTER 2
Local geometry of hypersurfaces
1
Lecture 11
Background from linear algebra: A symmetric bilinear form on Rn is a
map I : Rn Rn R of the form I (x, y ) = ij xi aij yj , where aij = aji .
Equivalently, I (x, y ) = x, Ay , where A is a symmetric m
CHAPTER 3
Global geometry of hypersurfaces
1
Lecture 24
Definition 24.1. A hypersurface is a subset M Rn+1 with the following
property. For every y M there is an open subset V Rn+1 containing y ,
and a function : V R whose zero set 1 (0) is precisely M V
18.950 Homework 2
1. (5 points) Let c : I R2 be a regular curve, whose curvature satis
es |c | 1 everywhere. Now stretch it in one direction, dening d(t) =
(2c1 (t), c2 (t). What can one say about d ? Obviously, the circle is the rst
example to think abou
18.950 Homework 3
Problem 1. (3 points) Write down explicitly a curve c : [0, ) R2 such
that the curvature (t) goes to innity as t .
Problem 2. (7 points) Let c : R R2 be a closed curve of period 5.
Suppose that it also satises
cos sin
c(t + 1) =
c(t),
s
18.950 Homework 10
1. (6 points) Check the formula for the geodesic equations on surfaces of
rotation from lecture 32.
2. (6 points) As before, consider a surface of rotation. Given c : I R2 ,
dene the angular momentum to be = l1 (c)2 c2 . Prove that if =
18.950 Homework 6
1. (6 points) Work out in detail the hump reversal method discussed
in Wednesdays lecture; see also textbook p. 157, Remark 4.25(ii). Verify
that the resulting surfaces really have the same rst fundamental form, but
generally dierent sec
18.950 Homework 7
1. (10 points) Complete the proof of the existence of local normal coordinates
(Corollary 21.3). You can use the beginning sketched in the lecture (in which
case you need to summarize it for the benet of the grader), or roll your own.
2.
18.950 Homework 9
1. (4 points) Prove that a torus cant have Gauss curvature which is every
where 0 (you may use the answers to the second midterm).
2. (6 points) Take M = S 2 to be the standard sphere. Find explicitly a
moving frame with singularities on
18.950 Homework 8
1. (10 points) Let M Rn+1 be a compact hypersurface, and p Rn+1 \ M
a point. The winding number is dened to be the degree of the map
M S n ,
y
yp
.
y p
Prove the following: if the winding number is nonzero, every smooth path
c : [0, )
18.950 Homework 5
1. (10 points) Let f be a hypersurface patch. Suppose that f lies in the
half-plane cfw_yn+1 0 Rn+1 , and that f is tangent to the hyperplane
cfw_yn+1 = 0 at x = 0. Prove that then, the principal curvatures at x = 0
satisfy i j 0 for all
18.950 Homework 1
1. (5 points) How does the curvature of a regular curve change under the
following transformations of the plane R2 : (a) translation, (b) rotation, (c)
reection, (d) dilation x rx?
2. (5 points) Compute the curvature of an ellipse. Where
18.950 Homework 4
Problem 1. Prove that 2 (L) vanishes if and only if L has rank 1.
Problem 2. Let L : R3 R3 be an invertible linear map. Prove that with
a suitable choice of basis of 2 (R3 ), the map 2 (L) : 2 (R3 ) 2 (R3 ) turns
into (L1 )tr det(L).
Pro