Math 6170              List of homework problems.              Fall 2005 
Treibergs

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HW 1.                                              Target Date: Sept. 12

This is problem 3, p. 9 of Sakai. Let  X, Y  be smooth vector fields on
the smooth manifold  M.  Let  h  be a smooth functon.

0.)  Show that  Xh = L_X h = dh(X)  (Lie derivative).

1.)  Show that 

             [X,Y] = L_X Y     [ = d/dt|_{t=0} df_{-t}(Y(f_t(p))) ].

     where  f_t  is the (local) flow generated by  X .
2.)  Let  g_s  be the (local) flow generated by  Y. Show that  [X,Y]  
     vanishes identically iff 
  
                     f_t  o  g_s  = g_s  o  f_t.

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HW 2.                                              Target Date: Sept. 14

Prove Cartan's Lemma: suppose {w_1,w_2,...,w_k}  is a set of smooth one-
forms of  M^n  which are independent at all points of  M. Suppose that
the one-forms {a_1,a_2,...,a_k}  have the property that 

            k
          \sum  a_i ^ w_i   =  0,
           i=1

where "^"  is wedge product.  Show that there are smooth functions
h_{ij}  such that  h_{ij} = h_{ji}  and that for all   i,

                           k
               a_i    =  \sum  h_{ij} w_j.
                          j=1

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HW 3.                                              Target Date: Sept. 19

Prove that the following models of the hyperbolic plane are isometric by
constructing maps that exhibit the isometry. (The maps may be realized 
geometrically!)

A. Hyperboloid model.  (X,g)
   
   X = { (x,y,z) \in R^3 : x^2 + y^2 - z^2 = -1, z > 0 }

   g = restriction of Minkowski metric ds^2 = dx^2 + dy^2 - dz^2  to  X

   (The geodesics turn out to be the intersections of  X  with planes 
   through the origin.)


B. Upper halfplane model.  (X,g)
   
   X = { (x,y) \in R^2 : y > 0 }

   g = y^(-2} ( dx^2 + dy^2 )

   (The geodesics turn out to be vertical lines and Euclidean
   semicircles centered on the  x-axis.)


C. Poincare model.  (X,g)
   
   X = { (x,y) \in R^2 : x^2 + y^2 < 1 }

   g = 4 (1 - x^2 - y_2 )^(-2) ( dx^2 + dy^2 )

   (The geodesics turn out to be lines through the origin and segments
   of Euclidean circles that meet  x^2 + y^2 = 1  perpendicularly.)


D. Klein model.  (X,g)
   
   X = { (x,y) \in R^2 : x^2 + y^2 < 1 }

   g = (1 - x^2 - y_2)^(-2) [ (1-y^2) dx^2 + 2xy dxdy + (1-x^2) dy^2 ]

   (The geodesics turn out to be Euclidean line segments that meet X.)

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HW 4.                                              Target Date: Sept. 21

Find the autoparallel curves in the Riemannian manifold

( R^2,  ds^2  =  ( 1 + y^2 )^2 dx^2 + dy^2    ).

You may wish to do instead

( R^2,  ds^2  =  exp(2y) dx^2 + dy^2    ).


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HW 5.                                              Target Date: Sept. 26

The Riemann curvature tensor in a local orthonormal coframe {w^i}
is given by

   \Omega_i^j = -1/2  \sum  R_i^j_p_q  w^p ^ w^q .

If the manifold is  n-dimensional, let  N(n)  be the number (dimension)
of independent components of the curvature tensor  R_i^j_p_q(x)  that 
can occur. Find N(n). (e.g.,  N(2)=1,  N(3)=6,  N(4)=20,....   )
[Christoffel, 1869.]

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HW 6.                                              Target Date: Sept. 28

Let  f:[0,1]x[0,a] -> M be a smooth map to a Riemannian manifold. 
Suppose that for all fixed  t_0 \in [0,a],  the curve  s -> f(s,t_0), 
s  in  [0,1]  is a geodesic parameterized by arclength, which is 
orthogonal to the curve  t -> f(0,t)  at the point  f(0,t_0).  Prove
that for all  (s_0,t_0) in  [0,1]x[0,a],  the curves  s -> f(s,t_0),   
and  t -> f(s_0,t)   are orthogonal.  [do Carmo, p. 84]


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HW 7.                                              Target Date: Oct. 3

(Text problem  78[II.5.i,iii].)  Let  (r,§)  be ordinary polar
coordinates in  R^2.  Suppose the metric  g  satisfies

     g( d/dr, d/dr ) = 1,

     g( d/dr, d/d§ ) = 0,

     g( d/d§, d/d§ ) = f(r)^2,  

     where f \in C^2, f(r,§)>0 for r>0, f(0,§)=0,  d/dr f(0,§)=1.

A.   Show that the radial curves  t -> (t,§)  are minimizing.

C.   Find the Gauss Curvature  K.

D.   Let  c \in R  be constant. Find such  f  so that  K = c  for 
     all  (r,§).

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HW 8.                                              Target Date: Oct. 5

Suppose that  (M^n,g)  is a Riemanninan manifold whose Riemannian
curvature, Ricci curature and scalar curvature are  R_i^j_k_l, R_ij and
R, resp.  Let  h = exp(2u) g  be a conformal metric, where  u  is a
smooth function on  M.  Compute the Riemannian curvature, Ricci
curature and scalar curvature  of  h  in terms of the curvatures of  g,
R_i^j_k_l, R_ij, R,  and  the function  u  and its first and second
derivatives. (The  n=2  case is considerably easier!)
[Problem from the text, p. 50.]


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HW 9.                                              Target Date: Oct. 10

Suppose that  (M^n,g)  is an oriented Riemanninan manifold. Define an 
n-form, the signed volume form (up to sign) by

        vol(V_1,V_2,...,V_n) =  \sqrt( \det( g(V_i,V_j) ) )

A. Suppose  {w^i}  is a local orthonormal frame. Show that, up to sign

        vol = w^1 ^ w^2 ^ ... ^ w^n

B. For a vector field  X  on  M, and any  k-form  §,  define the 
   interior product  i(X)§  as the k-1 form given by

        (i(X)§)(V_2,...V_k) = §(X,V_2,...V_k).

   For a vector field  X = \xi^i e_i, define the divergence, the 
   function  div(X) by

         div(X)  =  \sum \xi^i_,i  where   D_{e_k} X = \xi^j_,k e_j.

   Show that  d( i(X) vol ) = (dix X) vol.

C. For a function  f,  show that the Laplacian,

          \Delta f = div( grad f ).

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HW 10.                                             Target Date: Oct. 12

Suppose that  (M^n,g)  and (N^n,h) are two Riemannian manifolds whose
curvatures are  R  and  R' resp.  If  f:M -> N  is a diffeomeorphism
such that  f  preserves the curvature, i.e.,  for all  p  in  M  and
vectors X, Y, Z, W  at  p,

g< R(X,Y)Z, W >_p = h< R'(df(X),df(Y))df(Z), df(W) >_f(p),

does it follow that  f  is an isometry?  What if  R \ne 0?

Hint: See do Carmo, Riemannian Geometry, p. 59, where he gives the 
following picture.
       ________________    f    ______________________________
      (________________) ----> (______________________________)
       
See also do Carmo, Differential Geometry of Curves and Surfaces, p237
where he offers the surfaces in three space

        X(u,v) = (u cos v, u sin v, log u),
        Y(u,v) = (u cos v, u sin v, v).


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HW 11.                                             Target Date: Oct. 17

Let  (M,g)  be a complete, noncompact, connected Riemannian manifold and
let  p  be a point of  M.  Show that there is a geodesic ray emanating 
from  p.   That is, there is a unit speed curve  c:[0,\infty) -> M  such
that  c(0)=p  and for all  t>0,

               dist(p, c(t))  =  t.   


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HW 12.                                             Target Date: Oct. 19

Let  f:M --> (N,h)  be a smooth map between Riemannian manifolds.

a.) If  f  is a covering map, then there is a metric  g  for  M  so that
    f  is a local isometry.  In this case,  (M,g)  is complete if and 
    only if  (N,h)  is complete.

b.) But if  f  is just a local diffeomorphism, then there still is a 
    metric  g  so that  f  is a local isometry but this time it may be 
    that (N,h) is complete and (M,g) is incomplete. [do Carmo p. 152.]

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HW 13.                                             Target Date: Oct. 24

If  (M,g)  is a complete Riemannian manifold with nonpositive sectional 
curvaure, then every homotopy class of paths with fixed endpoints  p  
and  q  contains a unique geodesic. [Gallot, Hulin, Lafontaine, 
Riemannian Geometry, 2nd ed. p.138]

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HW 14.                                             Target Date: Oct. 26

If  M  is a Riemannian manifold possessing an isometry  f:M->M,  then 
any connected component of the set of all fixed points of  f  is totally
geodesic.

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HW 15.                                             Target Date: Oct. 31

Let  M  be a complete Riemannian manifold and  N  be a closed (not 
necessarily compact) submanifold of  M.  Let  p  be any point of  M  not
in  N.  Show that there is a point  q  in  N  so that the distance  
d(p,N)=d(p,q).  Show that if  c  is a minimizing geodesic from p to q  
then c  is orthogonal to  N  at  q. [do Carmo p. 207]

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HW 16.                                             Target Date: Nov. 2

Let  D  be a domain with smooth boundary in a Riemannian manifold and 
N  the outward normal vector field to the boundary  S  of  D.  For a 
smooth function  u  on  S  with compct support, set

    a(t,p) = exp_p (tu(p)N(p))

and

    S_t = { a(t,p) : p  in  S}.

Then  S_t  is a variation of  S. Let  D_t  be the domains bounded by  
S_t  so that  D_0=D.  Prove that

     d |                   /
    -- |    vol(D_t)  =    |   u dA
    dt |t=0                /S 

     d |                         /
    -- |    vol_{n-1}(S_t)  =    |  (n-1)Hu dA
    dt |t=0                      /S

where  H  is the mean curvature of  S. [Sakai, p. 130]


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HW 15.                                              Target Date:  Nov. 7

Let  M^n  be an orientable Riemannian manifold of even dimension  n  and 
positive sectional curvature.  Suppose  c  is a closed geodesic.  Then 
c  is homotopic to a closed curve whose length is strictly less than 
the length of  c. [do Carmo p. 208]

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HW 16.                                              Target Date:  Nov. 9

Let  M^n  be a Riemannian manifold of positive sectional curvature. 
Suppose  P  and  Q  are two totally geodesic submanifolds. Show that 
P  and  Q  intersect if
      dim(P) + dim(Q)  \ge  n-1
[This is a theorem of T. Frankel. See Petersen, Riemannian geometry, 
p. 162]

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HW 17.                                              Target Date: Nov. 14

Let  M   be a compact Riemannian manifold without boundary. Then every
nontrivial free homotopy class of loops contains a minimal length member
which is a smooth closed geodesic.

Hint. Let  s  be a loop (continuous map  s:S^1 -> M) in the nontrivial 
free homotopy class  A.  Approximate  s  uniformly by a broken geodesic 
loop  c_1  in  A.  Assume that the breaks are closer together than the  
injectivity radius of  M.  Then construct a sequence of broken geodesics 
inductively by letting  c_i  be the broken geodesic loop which has as 
its segments minimizing segments between the midpoints of the segments  
of  c_{i-1}.  Then  c_i  is in  A  and  c_i tends to a smooth closed 
geodesic of  A. [Bishop & Crittenden, Geometry of Manifolds, p. 243.] 

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HW 18.                                              Target Date:  Nov.16

A "line" in a complete Riemannian manifold is a geodesic  

      c: (-\infty, \infty) --> M

which minimizes length between any two of its points.  Show that if the  
sectional curvature is strictly positive then  M  does not have any  
lines. Give an example to show that this is false if  K \ge 0.
[do Carmo,  p. 252.]

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HW 19.                                              Target Date:  Nov.21

[Sakai, p. 107, Ex. 4.]  Suppose  M  is a compact Riemannian manifold
with dimension at least two. Show that  M  is simply connected if and 
only if its cut locus  C_p  is simply connected.

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HW 20.                                              Target Date:  Nov.28

Let  M  be a complete manifold of nonpositive sectional curvature and
Y(t)  a Jacobi field along a unit speed geodesic  c:[0,a] --> M  such
that  Y(0) = 0.  Show that for all  t>0,

         t |Y'(0)|  \le  |Y(t)|.

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HW 21.                                              Target Date:  Nov.30

Let  K  be a nonempty open set in  E^3  such that  the boundary is a
smooth surface.  For  r \ge 0,  let the parallel body be given by

     K_r  =  { x in  E^3 :  d(x,K) \le r }.

Let  S_r  denote its boundary surface. Find the second fundamental form
of  S_r.  Find  Area(S_r)  and  Vol(K_r),  all in terms of  K and r.
What are the formulas if  K  is a subset of a space form of constant 
sectional curvature?
(These are called  Steiner's formulas. For a convex domain  D  in E^2, 
for example,  the length of the boundary of  D_r  is  L+2\pi r  where  L 
is the length of the boundary of  D, and  

      Area(D_r) = Area(D) + rL + \pi r^2.)

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HW 22 .                                              Target Date:  Dec.5

Let  M  be a compact, n-dimensional manifold whose Ricci curvature is 
bounded below by  (n-1)k.  A subset  P  in  M  is said to be  
q-separated if any two distinct points in  P  have a distance at 
least  q.  If  P  is a maximal  q-separated set, show that

                       vol(M)
       card P  \ge  ------------
                     vol_k(B_q)  .

Here, vol(M) is the volume of  M  and vol_k(B_q)  is the volume of a  
q-ball in the space form of curvature  k. [Chavel, p. 169]


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HW 23 .                                              Target Date:  Dec.7

Let  M  be a complete, noncompact manifold of nonnegative Ricci 
curvature. Then there is a constant  0 < c_1(M)  so that

1.)  (R. Bishop) 
          vol(B(r,p)) \le  \vol_{E^n}(B(1,p)) r^n  for all  r > 0;

2.)  (S. T. Yau)  
                  c_1 r  \le  vol(B(r,p))    for all  r > 1.


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HW 24 .                                              Target Date: Dec.12

[Sakai, p. 238, Ex. 2.]  let  M  be a compact Riemannian manifold and  
pr:M --> M~  its universal cover. Lifting the Riemanniam metric, the
fundamental group acts by deck transformation isometries.  for p  in  M,
let  p~  in pr^(-1)(p)  and define a norm on the fundamental group by
||g|| = d(p~,g(p~)).  For a positive number  r, define

     N(r) = #{g in \pi_1(M,p): ||g|| < r}

     h(\pi_1(M,p)) = lim inf_{r\to\infty} log N(r) / r

     h(M) = lim inf_{r\to\infty} log(vol(B(r,p~)) / r.

Prove that
1.)  h(\pi_1(M,p))  and  h(M)  do not depend on the choice of  p~.
2.)  h(M) \ge h(\pi_1(M,p)).


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HW 25.                                              Target Date: Dec.14

Suppose  M^n  is a compact manifold whose sectional curvature is greater 
than  L>0.  Assume that the diameter 

     \pi / (2Ã L) <  diam(M)  \le \pi / \sqrt(L).

then there is a point  q  in  M  so that any nontrivial geodesic from  q
to  q  has length at least  \pi/sqrt(L).

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