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!
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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!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
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