Math 6170 List of homework problems. Fall 2007 Treibergs ------------------------------------------------------------------------ HW 1. Target Date: Aug. 30 Suppose that a hypersurface in Euclidean space M^n immersed in E^{d+1} is given locally as a graph of a smooth function defined over an open set U in R^d. X(u_1, u_2, ..., u_d) = (u_1, u_2, ..., u_d, f(u_1,u_2,...,u_d) ) Define the metric in these coordinates by / d \ / d \ g[i][j] = |----- X(u_1,...,u_d)| |----- X(u_1,..., u_d)| \ du_i / \ du_j / a.) Let c : [a,b] -> M be a piecewise C^1 curve in M. Show that its length as a curve in Euclidean space agrees with its length computed using the metric /b / | | d | | | |--- c(t)| dt = | ds | | dt | | /a /c where (1/2) / / d \ / d \\ ds = |g[i][j] |--- u_i| |--- u_j|| dt \ \ dt / \ dt // and c(t) = X(u_1(t),...u_d(t)). b.) Let K in U be a compact region with smooth boundary. Show that the n-dimensional area of X(K) computed as the area of a graph agrees with the area computed intrinsically using the area form: / (1/2) | / 2 2\ / | | / d \ / d \ | | | |1 + |----- f| + ... + |----- f| | du_1 ... du_d = | dA / \ \ du_1 / \ du_d / / | K /X(K) where (1/2) dA = det(g[i][j]) du_1 ... du_d. ------------------------------------------------------------------------ HW 2. Target Due Date: Sept. 6 Let G be a Lie group. (G is both a smooth differentiable manifold and a group whose multiplications and inverses are smooth operations.) Let G_e be the tangent space at the identity. For a fixed h in G, the left translation given by L_h(k) = hk is a diffeomorphism because (L_h)^{-1} = L_{h^{-1}}. A vector field X on G is left invariant if for all k in G, (d L_h X)(k) = X(hk). Let lie(G) denote the left invariant vector fields on G. lie(G) is isomorphic to G_e. It is a fact that if X, Y are in lie(G) then so is their bracket [X,Y] = XY - YX. Hence lie(G) is a Lie-subalgebra of the Lie-algebra of all vector fields on G. A one parameter subgtoup of G is a Lie-group morphism F: R -> G of the additive Lie-group of the real numbers R into G such that dF is nonzero. Hence F(s + t) = F(s) F(t), F(0 + t) = F(0) F(t) so F(0) = e. Define a connection LD by the condition LD_X Y = 0 everywhere on G for all X and left invariant Y. Since every vector field is a linear combination of left invariant vector fields with function coefficients, this condition determines the connection completely. Show that in G with the left invariant connection, the integral curves of the left invariant vectors are the auto-parallel curves with respect to LD such that their maximal domain of definition is the whole real line. Show also that the nonconstant auto-parallel curves of LD starting at e can be characterized as the one-parameter subgroups of G. ------------------------------------------------------------------------ HW 3. Target Date: Sept. 13 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 ------------------------------------------------------------------------ HW 4. Target Date: Sept. 17 Suppose that T(X,Y) is a skew symmetric tensor of type (1,2) on the smooth manifold M. Show that there is a unique metric compatible connection D on M whose torsion is T. Instead of formula I.5.4 of Chavel one gets 2< D_X Y, Z > = X + Y - Z + + - + + - . Suppose that T satisfies the additional condition =0 for all vector fields X, Y. Then the autoparallel curves for D coincide with the autoparallel curves of the Levi-Civita connection. [From Gromoll, Klingenberg & Meyer, Riemannsche Geometrie im Grossen, 2nd ed., p. 87.] ------------------------------------------------------------------------ HW 5. Target Date: Sept. 19 a.) Weaver's metric (also known as Chebychev Coordinates of a surface.) 2 Let phi(u,v) be a smooth function in R that satisfies 0 < phi(u, v) < pi. The metric 2 2 2 ds = du + 2 cos(phi) du dv + dv is known as the weaver's metric since length is preserved along the coordinate lines which correspond to the warp and weft threads, and phi is the angle between the threads. Find its Gauss curvature. b.) Stereographic Coordinates for the Sphere. n Find the Levi-Civita connection and curvature for the metric on R given by 2 4 / 2 2 2\ ds = ----------- \dx_1 + dx_2 + ... + dx_n / 2 / 2\ \1 + |x| / where 2 2 2 2 |x| = x_1 + x_2 + ... + x_n . ------------------------------------------------------------------------ HW 6. Target Date: Sept. 24 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.] ------------------------------------------------------------------------ HW 7. Target Date: Sept. 26 n Let f be a smooth function on the manifold M . The gradient vector field grad f is defined by X f = < grad f, X > for all X. Thus n grad f = \sum f_i e_i i=1 where {e_i} is an orthonormal frame for M and df = f_i w^i and where {w^i} is the corresponding orthonormal coframe. a.) Suppose that | grad f | = 1 everywhere. Show that the integral curves of grad f are auto-parallel. [Chavel I.4, p. 36.] The Hessian of f is defined as the two form gotten by covariant differentiation of the one form df ( Hess f )(X,Y) := D ( df ) (X,Y). Thus Hess f = f_{ij} w^i w^j where d f_i - f_j w_i{}^j = f_{ij} w^j. b.) Show that the Hessian is positive semidefinite at a local minimum of f. [Chavel 1.10, p. 41.] Let X be a vector field. The divergence is the function defined by n div X = trace(Y |-> D_Y X) = \sum < D_{e_i} X, e_i>. i=1 The Laplacian is defined n Æ f = div grad f = \sum f_{ii}. i=1 c.) Show div ( f X ) = < grad f, X > + f div X, and Æ (fh) = f Æ h + 2< grad f, grad h > + h Æ f. ------------------------------------------------------------------------ HW 8. Target Date: Oct. 3 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] ------------------------------------------------------------------------ HW 9. Target Date: Oct. 3 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. [Chavel, p. 37.] ------------------------------------------------------------------------ HW 10. Target Date: Oct. 17 Let G be any Lie Group with independent left invariant vector fields {e_1,...,e_n}, n = dim G, and dual 1-forms {w^1,...,w^n}. Show that there exist constants C_jk^i such that n n [e_j,e_k] = \sum C_jk^i e_i; d w^i = - (1/2) \sum C_jk^i w^j ^ w^k i=1 j,k=1 n \sum C_jk^m C_mi^r + C_ij^m C_mk^r + C_ki^m C_mj^r = 0. m=1 Assume G possesses a bi-invariant Riemannian metric relative to which {e_1,...,e_n} are orthonormal. Show C_ij^k + C_ik^j = 0. Show also that the connection forms relative to the frame {e_1,...,e_n} are given by n w_i^j = - (1/2) \sum C_ki^j w^k; k=1 so for left-invariant vector fields X, Y on G we have D_X Y = (1/2) [X,Y]. so R(X,Y)Z = (1/4) { [[Z,Y],X] - [[Z,X],Y] } and < R(X,Y)X, Y > = (1/4) |[X,Y]|^2 for all X,Y,Z in \frak g. [Chavel, p. 50.] ------------------------------------------------------------------------ HW 11. Target Date: Oct. 22 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.) E. Hemisphere model. (X,g) X = { (x,y,z) \in R^3 : x^2 + y^2 + z^2 = 1 and z > 0 } g = restriction of upper halfspace meric ds^2 = z^{-2} ( dx^2 + dy^2 + dz^2 ) to X (The geodesics turn out to be semicircles in vertical planes that meet X.) [For a wonderful treatment see J. Cannon, W. Floyd, R. Kenyon, W. Parry "Hyperbolic Geometry," in Flavors of Geometry, Silvio Levy, ed., MSRI Publications, 31, 59-116, Cambridge University Press, Cambridge 1997.] ------------------------------------------------------------------------ HW 11. Target Date: Oct. 24 a.) Let M be a submanifold of N. We say that M is totally geodesic in N if whenever there is a geodesic c:[a,b] -> N and a point t in (a,b) where c(t) is in M and c'(t) is tangent to M, then c is contained in M for a whole interval: there is a d > 0 so that c((t-d,t+d)) lies entirely in M. Show that M is totally geodesic if and only if the second fundamental form of M vanishes on M. b.) Show that if N is a manifold possesing an isometry I: N -> N, then any connected component of the set of all points left fixed by I is totally geodesic. [Chavel, p. 96.] ------------------------------------------------------------------------ HW 12. Target Date: Oct. 29 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] ------------------------------------------------------------------------ HW 13. Target Date: Oct. 31 Let M^n be a complete, orientable Riemannian manifold of even dimension n and uniformly 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] (Combining this with the fact that every nontrivial free homotopy class of M contains a closed minimizing geodesic shows that such manifolds are simply connected.) ------------------------------------------------------------------------ HW 14. Target Date: Nov. 7 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] ------------------------------------------------------------------------ HW 15. Target Date: Nov. 7 A pole is a point in a complete Riemannian manifold with the property that it has no conjugate points. Show that the point p = (0,0,0) of the paraboloid S = { (x,y,z) \in R^3 : z = x^2 + y^2 } is a pole of S and, nevertheless, the curvature of S is positive. [do Carmo, p. 154.] ------------------------------------------------------------------------ HW 16. Target Date: Nov. 14 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). ------------------------------------------------------------------------ HW 17. Target Date: Nov. 19 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.] ------------------------------------------------------------------------ HW 18. Target Date: Nov.26 Let M be a complete manifold of nonnegative sectional curvature. Let w, c : [0, \infty) be unit speed autoparallel curves emanating from p. Suppose that c is a ray: d(c(t),c(0)) = t for all t > 0. Suppose that the angle from c'(0) to w'(0) is strictly smaller that \pi/2. Show that lim d( w(s), w(0) ) = \infty. s -> \infty [Sakai, p. 190.] ------------------------------------------------------------------------ HW 19. Target Date: Nov.28 Let M be a complete, simply connected Riemannian manifold whose sectional curvature satisfies K \le 1. Suppose that for all p in M, the locus C(p) of first conjugate points to p reduces to a unique point other than p and that d(p,C(p)) = \pi. Prove that M is isometric to the sphere with constant cxurvature 1. [do Carmo, p. 251.] ------------------------------------------------------------------------ HW 20. Target Date: Nov.28 Prove the law of cosines and the law of sines for the space of constant curvature k [see Chavel p. 103.] Show that as k -> 0 the formulas limit to the Euclidean versions. e.g., if k = -1 then for the triangle with vertix angles A, B, C and opposite side lengths a, b, c, there holds cosh c = cosh a cosh b - sinh a sinh b cos C sinh a : sinh b : sinh c = sin A : ain B : sin C [Hint: put one of the vertices at (0,0,1) in the sphere or hyperboloid. Then rotate space.] ------------------------------------------------------------------------ HW 21. Target Date: Dec. 5 Let (M,g) be a compact Riemannian manifold and (M~,g~) be its universal cover with lifted metric and p:M~ -> M the projection. The fundamental group \pi_1(M.x) can be identified with the deck transformation group G. Fix a point x~ \in M~ so that p(x~)=x and define a norm of a in G by || a || = d(x~,a(x~)). With the generating set S = { a \in G : || a || \le 3 diam(M) }. Define a second norm on G given by the word norm |a| = least number of generators of S whose product is a. Prove that diam(M) |a| \le ||a|| \e 3 diam(M) |a|. For positive r, let N(r) = #{ a \in G : || a || < r } and log(N(r)) log(vol(B~(r,x~)) h(G) = lim inf ---------; h(M) = lim inf -----------------. r --> \infty r r --> \infty r Show h(G) and h(M) are independent of the choice of x~ in M~. Show h(G) \le h(M). [Sakai, p. 237.] ------------------------------------------------------------------------ HW 22. Target Date: Dec. 5 Let c:[0,\infty) -> M be a ray in M. To every t in [0,\infty) associate the function b( . ;t) : M -> R given by b(x;t) = t - d(x,c(t)). Prove: a.) | b(x,t) - b(y,t) | \le d(x,y) for all t,x,y. b.) | b(t,x) | \le d(x,c(0)) all t,x. c.) t>s implies b(x;t) \ge b(x,s) all s,t,x. Therefore one can define the Busemann function of c E(x;c) = lim b(x,t) t->\infty d.) If s>0, let the geodesic c(t;s) = c(s+t). Show E(x; c( . ;s)) = E(x;c) - s all s,x. e.) Consider the union of balls subset of M, called the horoball H(c) = U { B(c(t),t) : t>0 }. Show that E(x;c) = s for all s>0, x in the boundary of H(c( . ;s)) f.) Let G(c) = M \ H(c). Suppose M has nonnegative sectional curvature. Show that G(c) is totally convex. (i.e., for any p,q in G(c), any autoparallel curve of M from p to q lies in G(c).) [Chavel, p 421.] ------------------------------------------------------------------------ HW 22. All homework for the semester is due at the final exam time, Tue., Dec. 11, 5:30 p.m. Prove the following theorem of E. Calabi and S. T. Yau. Suppose M is a complete, noncompact manifold of nonnegative Ricci curvature. Then for every p in M there is a constant c > 0 so that the volume of the ball about p satisfies asymptotically as r --> \infty, vol(B(p,r)) \ge c r. This estimate is sharp for the Riemannian product M = N^{n-1} x R. [Chavel, p. 165.] =================================end====================================