These are notes and problems from Rob Kusner's Fall 2006 graduate course on Manifolds (Math 703 at UMassAmherst, this material Copyleft* 2006). They will be discussed further in problem sessions most Wednesdays at 4:15PM in the GANG Lab (1535 LGRT) - bring your tea and cake down! ====================================================================== 09/06 Special Lecture: 3-manifolds according to Grisha Perelman In its popular "border patrol version" the Poincare' Conjecture asserts that a 3-dimensional alien which can deform itself to slip out of any lasso is actually just a deformed 3-ball. Thus will begin a PG-version of the Thurston Geometrization and the Poincare' Conjecture for 3-manifolds, working up from the situation for 1- and 2-manifolds. [The X-rated version: Any compact 3-manifold X decomposes into a connected sum of primes P; P futher decomposes along incompressible tori to give components Q which are Seifert-fibered or contain no further incompressible tori; each Q carries one of the 8 homogeneous geometries in dimension 3 (S^3, R^3, H^3, S^2 x R, H^2 x R, SL(2,R), Nil (Heisenberg) and Solv (dilations and translations of R^2). In case the fundamental group G of X is finite, this gives a spherical space-form S^3/G, with G represented in SO(4); in particular, for G trivial, the Poincare' Conjecture: X = S^3.] Using Ricci-flow, a diffusion process that averages a Riemannian metric according to how (Ricci) curved it is, and careful surgery arguments to deal with singularities in the flow, it appears Grisha Perelman has effectively completed Richard Hamilton's program to geometrize any compact 3-manifold! Here are some references at increasing levels of depth: "Structure of Three-Manifolds - Poincare' and Geometrization Conjectures" (text and figures from a summer 2006 lecture by Harvard geometric analyst and 1983 Fields Medalist Shing-Tung Yau, putting Perelman's work into context for a general audience - the context includes work by Yau and his collaborators and students, and may serve as an object lesson for the future): http://www.math.umass.edu/~kusner/yau_poincare.pdf [also at: arXiv:math.DG/0607821] * * * "Three-Manifolds According to Grisha Perelman" (a spring 2003 triptych by fellow UMass grad students Eli Cooper and So Okada - and yours truly - summarizing Perelman's first public lecture on his solution of the Geometrization and Poincare' Conjectures via Ricci flow): http://www.gang.umass.edu/~kusner/other/new-perelman.pdf * * * "Recent Progress on the Poincare' Conjecture and the Classification of 3-Manifolds" (a fall 2004 survey printed in the Bulletin of the AMS, volume 42:1, pages 57-78, perhaps the most concise treatment of the math): http://www.ams.org/bull/2005-42-01/S0273-0979-04-01045-6/home.html * * * The website containing the most up-to-date notes and commentary for mathematicians on Geometrization Conjecture: http://www.math.lsa.umich.edu/~lott/ricciflow/perelman.htm 09/11 Discuss these differential structures (maximal atlases) on \R: the atlas \A, which contains the chart (\R, \phi(x) = x); and \B, containing the chart (\R, \psi(x) = x^3). Does \A = \B when we insist that these structures are smooth? C^k? C^0? How smooth is the differential structure on S^1 whose charts include stereographic projection from north and south poles? (Work this out explicitly!) What fails with the "foliations" example -- the "leaves" are integral curves of some ODE in \R^2: we saw the "leaf space" is locally homeomorphic to \R, but it isn't a 1-manifold? Can you give an even simpler example of this failure? [The historical roots of this formulation of a differential structure are found in Hermann Weyl's (1913) Idee des Riemannflache; 2e (1955) The Concept of a Riemann Surface. Addison-Wesley.] 9/13 Suppose M is \R with its usual smooth structure (the one compatible with the chart \phi(x)=x), and N is the one with chart \psi(x)=x^3. Show that M and N are diffeomorphic by constructing an explicit diffeomorphism f:M -> N. Can you show that any two smooth structures on \R are diffeomorphic? The Riemann sphere S^2 has a pair of complex valued charts z and w which satisfy z=1/w on the coordinate overlap region \C - \{0\}, so the change of coordinates is complex analytic. The torus S^1 \times S^1 can be given a real analytic structure as a product manifold. Can you find explicit charts to give the torus a complex analytic structure? The real projective space \RP^n has an atlas containing n+1 coordinate charts -- called affine charts -- where chart U_j is defined as the set of ratios [x_0:...:x_n] where x_j does NOT vanish. Work out the change of coordinates from U_j to U_k in terms of these x_j and x_k to see that this is a real analytic (in fact rational) structure on \RP^n. Generalize affine coordinate charts to the Grassmanian G_{k,l}(\R) of real linear k-planes in R^l and show it is a smooth k(l-k)-manifold. (Note that \RP^n = G_{1,n+1}(\R). The analog of the affine chart U_j, which consists of ratios which can scaled to have x_j = 1, will be a chart consisting of k by l matrices which have l-k rows "scaled" to be the basic unit vectors of \R^k -- there are l-choose-k of these charts.) Note that the last two examples work with any field in place of \R. In particular for \C. Compare charts for \CP^1 with those for the Riemann sphere S^2. Verify the details that exp:\R -> S^1 is a covering space, where exp(t) = e^{i2\pi t}. The integers \Z all map to the point 1, so the various levels (decks or "pancakes") are labeled by \Z, when one thinks of the picture with \R "wrapped around" S^1. Another picture has \Z translating \R in the usual way (by addition), and a fundamental domain of this covering space (the interval (0,1), for example) then "tiles" \R by these \Z translations (deck transformations). Explore the n-dimensional analog \R^n/\Z^n. Show that S^n -> \RP^n is a covering space with covering projection being the quotient which identifies a pair of unit vectors u and -u with the line between them. (Only two "pancakes" in this stack!) Warm-up for the inverse function theorem: If f:\R -> \R is smooth, and df/dt is never 0, then f is a diffeomorphism onto its image. Also prove the complex analog of this, for holomorphic f:\C -> \C. (Recall and verify that det(Df)=|df/dz|^2 to do this!) Problem session at 4:15PM in GANG Lab (1535 LGRT) 09/18 Check that a (regular) submanifold is a manifold of the appropriate dimension, and that the atlas defined by restricting charts is as smooth as the atlas of the ambient manifold. We noted that an m-dimensional linear subspace V of \R^n is a (regular) submanifold. Invesitigate the analogous submanifolds for S^n and \RP^n. The irrational line on the torus is an immersed submanifold, but not a (regular) submanifold, since it cannot be expressed locally as the zero set of a coordinate (say \{x_2 = 0\}) in chart. Show it can be locally expressed as a dense subset of the union over c \in \R of level sets of the form \{x_2 = c\}. In fact, this is an example which satisfies the formal definition of a foliation of a manifold. Try to formulate the concept of a foliation of an n-manifold, each of whose leaves is an m-manifold (the local picture will now look like \{x_{m+1} = c_1, ... x_n = c_{n-m}\}; as the c's vary, one goes from one leaf --- at least locally --- to another). 09/20 Earlier we said: an "n-manifold M with boundary \del M" has charts modeled on neighborhoods in the (closed) upper half space \R^n+, which have smooth change of coordinates on (slightly larger) open neighborhoods in \R^n. In the same spirit as the previous problem, check that the boundary \del M is an {n-1}-manifold, again as smooth as M. Furthermore, M can be "doubled" along \del M to get an "honest" n-manifold 2M which has \del M as a regular submanifold. What is the double of an interval? a 2-disk? an n-disk? an annulus? a 3-dimensional solid torus (bagel)? Show that the graph of a smooth map f:M -> N defines a (regular) submanifold of the product manifold M \times N. Problem session at 4:15PM in GANG Lab (1535 LGRT) 09/25 In case N = \R, try doing the preceding problem by expressing graph(f) as F^{-1}(q) for a regular value q of a map F: M \times \R -> \R. Can you make this argument work for general N? Verify that SL(n,\R) and O(n) are submanifolds of \R^{n \times n} by showing that 1 and I are regular values of the maps f(A) = det(A) and g(A) = A*A, repsectively, into \R and into \Sym(n,R). (Here A* is transpose of A, the n-by-n identity matrix is I, and \Sym(n,R) is the linear submanifold of symmetric n-by-n matrices.) Find a smooth function f:\R^3 -> \R which expresses a standard torus of revolution as the level set f^{-1}(t) of a regular value t. Can express surfaces of higher genus as a submanifold of R^3 similarly? (Hint: think of distance function from an appropriate plane curve.) Analytic inverse/implicit function theorem: If Df|_p is nonsingular, and f is analytic in a neighborhood of p, the local inverse f^{-1} not only is as smooth as the original f, but also has a convergent power series expansion on a (possibly smaller) neighborhood of f(p). 09/27 Special Lecture (Mike Sullivan): Prove the theorem which we stated: The tangent space TM|p of a manifold M at p, viewed as the space of derivations, has basis given by the first order parial differential operators \d/\dx_i (\d_x_i for short), where (x_1, ... , x_m) is a coordinate chart centered at p. What happens if you pick a different chart (y_1, ... , y_n) -- what is the expression for the d_y_j in terms of the d_x_i? Let f: M -> N be smooth map with f(p) = q. Then there is a linear map Tf|p which carries TM|p to TN|q whose matrix (with respect to the bases suggested in the previous problem) is Df|p. Can you define this linear map Tf|p directly using derivations (or equivalence classes of curves through p and q), check that it is linear, and that it has matrix Df|p when coordinate charts are chosen? Compare with the situation in linear algebra: what happens when you change coordinates on M and N? Suppose a smooth curve c:\R -> M satisfies c(0) = p, and that x_1, ... , x_n are local coordinates on M around p. Then c(t) is determined by x_1(t), ... , x_n(t) and its derivative c' = x_1'(0) \d/\dx_1 + ... + x_n'(0) \d/dx_n is a tangent vector, in the sense of derivation on smooth functions (at least those restricted to a neighborhood of p) on M. Conversely, decide when a pair of curves through p determine the same tangent vector. 10/02 Special Lecture (Mike Sullivan): Give an example of a non-vanishing tangent vector field on S^1. For any M, show that S^1 \times M has a non-vanishing smooth vector field. In particular, a "hairy" 2-torus T^2 = S^1 \times S^1 can be "combed". We claimed that you cannot comb a hairy S^2, but what about S^n for larger n? Can you find a non-vanishing vector field on S^3? on any odd dimensional sphere? We say that a manifold M is k-parallelizable if there are smooth vector fields V_1, ... , V_k such that at every point p of M, the k tangent vectors V_1(p), ... , V_k(p) are linearly independent in TM|p. Clearly k is at most dim M, and k is at least 1 if it can be combed. Show that the n-torus T^n is n-parallelizable. Also show S^3 is 3-parallelizable (the same is true for any orientable 3-manifold, but this is harder to prove). Suppose M^m is codimension-k submanifold of \R^n (m+k=n) which is expressed as the level set of a smooth map G: R^n -> R^k. Then the tangent spaces TM|p can be viewed as both the kernel of TG|p, and as the image of TJ|p under inclusion map J: M -> R^n. 10/04 Special Lecture (Mike Sullivan): 10/09 NO CLASS [to atone for what Columbus did to the Arawaks in 1492] 10/11 Problem session at 4:15PM in GANG Lab (1535 LGRT) 10/16 The Klein bottle K is the double of the M\"obius band. Show that K is a circle (non-product) S^1-bundle over S^1. (In fact, K and the 2-torus are the only compact 2-manifolds which are S^1-bundles, or bundles over S^1. There are many more 3-manifolds which can be constructed as surface bundles over S^1.) Show that S^3 is an S^1-bundle over S^2. More generally, show that S^{2n+1} is an S^1-bundle over \CP^n (the Hopf fibration). Try to state and prove analogous results for S^0 =\{+1,-1\} and S^3 in place of S^1, and with \R and \H in place of \C. 10/18 Calculus revisited: A section s: \R -> E of the trivial bundle E = \R \times \R is equivalent to a function f: \R -> R; a section of the bundle E' = \S^1 \times \R is equivalent to a periodic function; what about a section of the M\"obius band, i.e. the nontrivial \R-bundle E" over S^1? A rank n vector bundle is trivial iff it admits a global n-frame (compare with parallelizability). Recall the atlases for the examples M = S^n, T^n, \RP^n, \CP^n, a Riemann surface, etc. Work out the transition functions for the local coordinate trivializations of the tangent bundle TM for these examples. (A good example to start with is the Riemann surface S^2 = \CP^1 with the charts z = 1/w!) Problem session at 4:15PM in GANG Lab (1535 LGRT) 10/23 If vector bundles over M are trivial, so is their direct sum. What about the converse? (We already saw that a trivial bundle and a nontrivial bundle can sum to a trivial one, e.g., with the normal and tangent bundles of S^2 in \R^3.) Give some examples. 10/25 Work out the transition functions for the tautological line bundle L -> \RP^n, whose fibre over p is the line in \R^{n+1} "through" p. In case n=1, is L the annulus or the M\"obius band? (Study this construction also for \F = \C or \H.) Similarly, one can define the tautological vector bundle of rank k over the Grassmannian G_{k,n} of k-planes in \F^n.... Discuss various isomorphisms between vector bundles: E and its dual E* = Hom(E,\R); E \tensor F and Hom(E*,F), etc. Show that the orbit of a flow (an \R-action) on M is either a point or an immersed curve. Must the curve be (properly) embedded in M? Problem session at 4:15PM in GANG Lab (1535 LGRT) 10/30 What are the orbits for the flows on \R^2 defined by vector fields X_1 = x\d/\dy-y\d/\dx, X_2 = x\d/\dx+y\d/\dy = r\d/\dr, and X_3 = \d/\dr ? Which of these have defined for all \R? If not all \R, what is the maximal subinterval of \R on which they are defined, and how does it depend on the initial condition? Don't get into any mischief tonight, and Happy Hallowe'en!!! 11/1 The vector field X_3 above is not continuous on \R^2, but is on the punctured plane. As we observed in class, X_3 is not complete. For what \a is X = r^\a X_3 complete (we observed in class that X_2 is complete, so \a = 1 works - find all such \a). Give examples of vector fields Y_1 or Y_2 on S^2 with exactly 2 zeros whose flow has 2 fixed points N and S, and all other orbits of the flow are 1) lines joining the points N and S, or 2) loops around the points N and S. Describe the orbits of Y_s = s Y_1 + Y_2 for various s. Find a vector field on S^2 with exactly 1 zero. Can you find one with any number of zeros, perhaps even with ZERO zeros?!?!?!?!? Can you find a vector field on the disk D^2 with no zeros? How about one with no zeros which also points inward along the boundary circle? What about on the annulus S^1 \times [0,1]? Suppose two vector fields X and Y on M satisfy [X,Y] = 0. Show that their flows F and G: \R times M -> M commute, that is, f_t g_s = g_s f_t for all s and t where defined. Generalize! Problem session at 4:15PM in GANG Lab (1535 LGRT) 11/06 A diffeomorphism f:M -> N induces a Lie algebra homomorphism f_*: Vec(M) -> Vec(N). (For general smooth maps, what can be said?) Verify that, up to isomorphism, the only simple 2-dimensional Lie algebra over \R has a basis \{e,f\} satisfying [e,f] = e. What are the simple 3-dimensional Lie algebras over \R? (Over \R there seem to be two versions of so(3), although over \C, this distinction disappears. What about sl(2)...?) 11/08 Explore the compact Lie groups O(n), U(n), SU(n) and Sp(n). (What are their dimensions? Describe their Lie algebras....) What is the Lie group whose Lie algebra is isomorphic to the 2-dimensional simple Lie algebra mentioned last time? Problem session at 4:15PM in GANG Lab (1535 LGRT) 11/13 The Lie groups we have considered are all subgroups of G = GL(n,\R), whose Lie algebra of left-invariant vector fields is identified with TG|_I = \R^{n\times n} (as vector spaces). Show that the Lie bracket [X,Y] of left-invariant vector fields on G is identified with the commutator of the corresponding matrices in \R^{n\times n}, so that this identification is actually a Lie algebra (not only a vector space) isomorphism. Show that on any group, the actions of left and right translation commute with each other. (What group law is this?!) In the past, I have blithely referred to the quaternionic vector space \H^n without carefully pointing out that the action of the scalars \H is normally taken to be on the right. With this in mind, show that GL(n,\H) can be regarded as the subgroup of GL(2n.\C) which commute with the matrix | O -I_n | J_n = | | | I_n O | by showing that J is the matrix for right multiplication by the quaternion j with respect to a \C-basis of \C^n + \C^n = \H^n where the first summand is the (1,i)-part and the second is the (j,k)-part. Similarly, we can also express GL(n,\H) as the subgroup of GL(4n,\R) which commute with J_2n and with the 4n x 4n matrix | i O | J'= | ... | | O i | whose 2n blocks are each the 2 x 2 matrix i = J_1. In particular, we can thus think of Sp(n) as the intersection of GL(n,\H) with O(4n), that is, as the isometries of \H^n, just as we think of U(n) as GL(n,\C) intersected with O(2n). Show that on any group, the actions of left and right translation commute with each other, (What group law is this?!) 11/15 Let X be the velocity vector field of the flow F associated with a 1-parameter subgroup g_t of a Lie group G, that is, suppose X(h) = d/dt(F(t,h)|_{t=0} (t\in\R, h\in G) where F(t,h) = h g_t. [N.B. the action of g_t is on the *right*!] Show that X is left-invariant. More generally, show that whenever G acts on a manifold M, and for any 1-parameter subgroup g_t of G, the flow on M induced by g_t defines a vector field X' on M which is the push-forward of a vector field on G \times M by the action.... (Explore how left invariant X on G pushes forward to X'....) Visualize left-invariant vector fields on G = S^3 by stereographic projection to \R^3 with 1 \in S^3 projecting to 0 \in \R^3. How do these compare with left-invariant vector fields on the abelian Lie group \R^3? Problem session at 4:15PM in GANG Lab (1535 LGRT) 11/20 Explore the exponential map for subgroups of GL(n,\F), where \F = \R, \C or \H, especially for the classical groups above. Generally, exp:\g -> G is a diffeomorphism from a neighborhood of 0 in \g onto a neighborhood of 1 in G. On such neighborhoods, it has (local) inverse, denoted log. Find a power series expression for log in the case of GL(n,\F), etc. The exponential map \R^n -> T^n is a covering map onto the n-torus. (Can you find a non-commutative example with this property?) But in general, exp need not be a covering map (nor even a local diffeomorphism), and though it must map *into* the 1-component of G, it need not map *onto* this component (try sl(2,\R) -> SL(2,\R) as an example: let N be the nilpotent 2 by 2 matrix with a 1 in the upper right; the matrix N-I has det(N-I)=1, and is in the 1-component (why?), but N-I is not exp(X) for any matrix X with tr(X)=0). Express elements of S^3 = Sp(1) = SU(2) as exp of pure imaginary quaternions or as exp of certain 2 by 2 matrices, and observe how various 1-parameter subgroups sit inside as S^1 = U(1). 11/22 MOUNTAIN DAY [take a hike in the hills before it snows! :] 11/27 A morphism of Lie groups is a smooth map f:H -> G which is also a group homomorphism. Its derivative df|_1 induces a Lie algebra homomorphism \f:\h -> \g. Show this commutes with the exponential maps for H and G. Conversely, a Lie algebra homomorphism induces a morphism of Lie groups, at least when H is simply-connected. Work this out for the example we discussed today in class, where \su(2) -> \so(3) via \sigma_x -> i\cross, etc. (Here \sigma_x is the Pauli spin matrix and i\cross is the 3 by 3 skew matrix for the action of cross product by i, etc. - I wrote these down in class, so I am not going to try to type them all in here.) You should find the morphism on Lie groups SU(2) -> SO(3) is 2-to-1, and corresponds to the conjugation action of SU(2) on 2 by 2 hermitian matrices. 11/29 One way to prove that a Lie algebra morphism gives rise to a group morphism (as above) is to show: 1) graph(f) is a Lie subalgebra of \h + \g, isomorphic to \h; 2) argue that this subalgebra of \h + \g gives rise to a subgroup H~ of H\times\G; 3) show projection of H~ to H is a covering map; and thus 4) H~ is the graph of some group morphism \f which necessarily commutes with exp's. Step 2) is an important result in its own right: a Lie subalgebra \h \subset \g gives rise to a (unique) Lie subgroup H \subset G. To prove this it is tempting to take H = exp(\h), but we have already seen that exp needn't be onto; so we need to do a little more. This leads us to study the Frobenius "integrability" condition (FIC). Show that if a manifold M is (smoothly) foliated by k-submanifolds (the leaves of the foliation), then the tangent k-planes of these leaves define a (smooth) rank k subbundle E of TM (often called a k-plane distribution on M), which is "integrable" in the following sense: if E is (locally) framed by tangent vector fields X_1,..., X_k on M, then their Lie brackets [X_i,X_j] also lie in E (that is, [E,E] \subset E. (This turns out to be sufficient as well: FIC!) Consider a 2-plane distribution E on \R^3 spanned by X_1 = \d/\dx and X_2 = a(x) \d/\dy + b(x) \d/\dz, where a(x) goes monotonically from -1 to +1, and where b(x) goes from 0 up to 1 and back to zero, as x goes from -infty to +\infty (this is a "contact" distribution). Does E satisfy FIC? Problem session at 4:15PM in GANG Lab (1535 LGRT) 12/04 We used linear algebra to reduce the FIC (Frobenius integrability condition: E is involutive: [E,E] \subset E) to the "flat case". Please verify that the local frame (Y_1,...,Y_k) for E defined by Y=BX (recall that B is the inverse of A, where A is the nonsingular k-by-k matrix expressing the projection of the original framing (X_1,...,X_k) of E onto the first k coordinate vector fields) is flat, i.e. that [Y_i,Y_j] = 0 for all i,j. (Hint: since E is involutive, [Y_i,Y_j] is a linear combination of {Y_1,...,Y_k}; now expand both sides, using the fact that coordinate fields commute, to see the trivial linear combinaton is the only one possible!) Let X_1, X_2 be linearly independent vector fields on (an open subset of) \R^2. Note that E = span{X_1, X_2} is involutive (why?)! Now please carry out explicitly the change of framing suggested above to make a flat framing by Y_1, Y_2. Suppose Y = (Y_1,...,Y_k) is a globally defined flat k-frame (linearly independent and all their brackets vanish) on a compact manifold M. Let E = span{Y} be the associated involutive k-plane distribution which by FIC gives a foliation \L of M. Every leaf L of \L has universal cover \R^k. What does this tell is when k = dim(M)? Let \t be a maximal abelian* subalgebra of \g = Lie(G), for a compact Lie group G. Then exp(\t) is abelian, and is of the form \R^k/lattice, a MAXIMAL TORUS in G. (Recall that a lattice in \R^k is a rank k abelian group whose generators are also a basis for \R^k.) [*This means all brackets vanish.] 12/06 The dimension of any maximal torus in G is denoted rank(G). Compute rank(G) for the classical groups G = SO(n), U(n), SU(n) and Sp(n). We defined a G-homogeneous space M as a manifold on which the Lie group G acts transitively. We can write M = G/G_p where G_p is the isotropy group of a point p in M. Show that the isotropy group is a closed subgroup of G. (Note: changing p to q gives a conjugate G_q, and it is common to denote any one of these subgroups by H.) Verify that the homogeneous space quotient map G -> G/H is a principal H-bundle. Also verify that the base is a smooth manifold with dim(G/H) = dim(G) - dim(H). In fact, the tangent space T(G/H)|H at the identity coset H is naturally the quotient of Lie algebras \g/\h. Discuss Ad:G\timesG -> G:(g,h) -> ghg^{-1}, Ad_*:G\times\g -> \g and ad = (Ad_*)_*:\g \times \g -> \g. At least in case of matrix groups, show that ad_X(Y) = [X,Y]. Describe \RP^n, \CP^n and \HP^n as homogeneous spaces. Generalize to the corresponding Grassmanians and Stiefel varieties. Problem session at 4:15PM in GANG Lab (1535 LGRT) 12/11 We have seen how to construct a Riemannian metric on a vector bundle E. In case E = TM, we call this a Riemannian metric on M. The length of a path [0,1] -> M is defined as the integral of the length of it's velocity vector. Show that infimizing this among paths joining points p and q in M defines a metric function d(p,q). Thus a Riemannian metric on M defines a metric space structure on M in the usual sense. Reduction of structure group: Let E -> M be a vector bundle, which is associated to the principal bundle GL(E) -> M of linear frames in E. (In effect, E and GL(E) have the same transition functions, with values in G = GL(r=rank(E),\R), except those for GL(E) act by left translation on the fibre G, while those for E act by the usual action of G on \R^r.) Now let E be endowed with any Riemannian metric. Then we will have an associated principal O(r) bundle O(E) of all orthonormal frames of E. The Gram-Schmidt process gives an explicit reduction from GL(E) to O(E), and shows that GL(E) is diffeomorphic (in fact, isomorphic as bundles over M) to O(E) \times \R^{r(r+1)/2}. (Here \R^{r(r+1)/2} = GL(r,\R)/O(r) is the space of positive upper triangular matrices.) Show that any Lie group G has a left- (or right-) invariant Riemannian metric, whose 1-parameter subgroups are (locally) shortest paths, i.e. GEODESICS in G. (Hint: Any section of TG is a linear combination of left-invariant vector fields with smooth coefficients.) Tubular neighborhoods: Let N be a (compact) embedded submanifold of M. Show that N has a neighborhood in M which is diffeomorphic to the normal bundle \perpN of N. One way to do this: Endow M with any Riemannian metric. Then \perpN will be the orthogonal complement to TN of TM|N. Argue that there is an \eps > 0 so that any point p of M with distance \delta = d(p,N) = d(p,q) < \eps can be joined to q\in N by a unique shortest path (geodesic, parametrized proportional to arclength) c:[0,1] -> M, with c(0)=q and c(1)=p. The initial tangent c'(0) will be a normal vector v in \perpN|q of length \delta. The correspondence v -> p defines the NORMAL EXPONENTIAL MAP from a neighborhood of the zero section exp_N:\perpN_{\eps} -> M. (Note: in case N = {q}, we simply get exp_{q}:TM|q_{\eps} -> M; please compare this with previous problem in case M = G and q = 1 to justify the name "exp"! This problem is awkward since we have yet to define geodesics using connections/parallel translation....) A smooth fibre bundle E -> M is a surjective submersion. In case M is compact, connected manifold, show the converse: Use implicit function to show each fibre is smooth, and in fact has trivial normal bundle (hint: the differential of the submersion can be used to define a global framing of the normal bundle to each fibre). Now use the result of the previous problem to construct local trivializations. ******Topical talks in lieu of your final exams!!!****** 12/13 Existence of partitions of 1 (Aaron); Tensor fields and Lie derivatives (Diego); 12/14 The Clasical Lie groups (Greg); Adjoint action of a Lie group and the Weyl group (Garret) 12/15 Linking number and knot invariants (Nate); Morse theory applied to real problems (Louis); TBA (Shigeki) Have a wonderful winter break - I look forward to seeing you in the spring!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! All the material on this website is Copyleft* by Rob Kusner. *"Copyleft" means that you have permission to use this material for non-commercial purposes as long as you acknowledge its source. Any use, in whole or in part, must include the "Copyleft 2006 by Rob Kusner" message or its equivalent. Commercial use without prior approval of the copyleft holder is strictly forbidden, and is punishable by methods not even imagined at G't'mo!