Math 240B Spring 2000 Homework Assignements

Home Work # 6 due Friday, March 10

#1. If M is a complete Riemannian manifold and $N \subset M$ is a closed. embedded submanifold with the induced Riemannian metric, show that N is complete. [Warning: The distance function on N induced from the metric space structure of M is not in general equal to the Riemannian distance function of N.]

#2. Let f(x)= d(p,x) be the distance function at p. Show that on a geodesic polar coordinate neighborhood U of p (induced by $\exp_p$), grad $f = \frac{\partial}{\partial r}$ on $U-\{p\}$.

#3. Find the cut locus of a general flat torus.

#4. (Petersen) Page 161 #3

Home Work # 5 due Friday, March 3

1. Let $f_1, f_2: M \rightarrow M$ be isometries of a complete, connected Riemannian manifold M. Suppose that f₁(p) = f₂(p) and df₁ = df₂ on T_pM for some $p \in M$, show that $f_1 \equiv f_2$.

(Petersen) Page 135 #11

(do Carmo) Page 83 #7,9

Home Work # 4 due Friday, February 25

(Petersen) Page 134 #1,8(a),16

(do Carmo) Page 153 #9,10

Math 240B Spring 2000 Midterm due Wednesday, February 16

1. (Surface of Reverlution) Let $\gamma (t) = (x(t), y(t),0)$ in ${\mbox{\BBb R}}^3$ parametrized by arc length (i.e. $\vert\gamma'(t)\vert =1$) and y(t) > 0 for all t. By rotating this curve around the x-axis, we get a surface that can be represented as $(t,\theta) \rightarrow f(t, \theta) = (x(t), y(t)\cos \theta, y(t) \sin \theta)$.

a) Show that the induced metric on the surface can be written as $g = dt^2 + y^2 d\theta^2$.

b) Compute the sectional curvature of the surface and verify that when y(t) =t (the Euclidean plane), the sectional curvature $K \equiv 0$ and when $y(t) = \sin t$ (the unit sphere without north south poles), the sectional curvature $K \equiv 1$.

2. Consider the parametrizations of T² defined on $S^1\times S^1$ by

$$\phi_1 (\theta, \psi) = (\cos \theta, \sin \theta, \cos \psi, \sin \psi)$$

and

$$\phi_2 (\theta, \psi) = ((2+\cos \theta)\cos \psi, (2+\cos \theta)\sin \psi, \sin \theta).$$

The torus T² is hence equipped by two Riemannian submanifold structures (induced from ${\mbox{\BBb R}}^4$ and ${\mbox{\BBb R}}^3$ respectively). Compare in these two cases $[\frac{\partial}{\partial \theta}, \frac{\partial}{\partial \psi}]$ and $\nabla_{\frac{\partial}{\partial \theta}} \frac{\partial}{\partial \psi}$.

3. Suppose G is a Lie group with a bi-invariant metric and X, Y, Z are left invariant vector fields on G.

a) Show that the integral curve of a left invariant vector field X is a geodesic, i.e. $\nabla_X X = 0$. (Note that the integral curve of X gives a 1-parameter group action on G by right multiplication.)

b) Using a), show that $\nabla_X Y =\frac{1}{2} [X,Y]$, $R(X,Y)Z = \frac{1}{4} [Z,[X,Y]]$, and $K(\sigma) =\frac{1}{4} \Vert[X,Y]\Vert^2$ if X,Y is an orthonormal basis of the plane $\sigma$, concluding that the sectional curvatures are nonnegative.

Home Work # 3 due Friday, February 11

1. Show that if M³ is a connected Einstein manifold then M³ has constant sectional curvature.

2. (do Carmo) Page 106 #8, Page 58 #7:

(Page 106 #8. (Schur's Theorem) Let Mⁿ be a connected Riemannian manifold with $n \geq 3$. Suppose that M is isotropic, that is, for each $p \in M$, the sectional curvature $K(p,\sigma)$ doesn't depend on $\sigma \subset T_pM$. Prove that M has constant sectional curvature, that is, $K(p,\sigma)$ also doesn't depend on p.

Hint: Use the 2nd Bianchi identity.

3. (Page 58 #7. Let $S^2 \subset R^3$ be the unit sphere, c an arbitrary parallel of latitude on S² and V₀ a tengent vector to S² at a point of c. Describe geometrically the parallel transport of V₀ along c.

4. (Petersen) Page 85 #5

Home Work # 2 due Friday, February 4

(do Carmo) Page 57 #3,8a);
(Petersen) Page 54 #5,8,11,14,21,22,30

Optional (Petersen) Page 54 #6,7,

Home Work # 1 due Friday, January 21

Page 17 #1 (Petersen) (note the misprint: $\frac{1}{\sqrt{2}}$ should be $\frac{1}{2\pi}$);

#2 (Theorem 4.6, Boothby) Let M be a compact manifold. Use partition of unit to show that M can be embedded into some Euclidean space.
#3 Let $S^2 \subset {\mbox{\BBb R}}^3$ be the unit sphere with induced Euclidean metric. Write this metric of S² explicitly in terms of the spherical coordinate.