January
17 W preview (see posted notes)
follow MIT notes ("L *" indicates lecture number)
22 M a few words about the Klein bottle, higher genus surfaces
defined diff. manifolds, atlases, charts
atlases for the two-sphere (L 1)
24 W the projective space (L 1)
smooth maps, diffeomorphism (L 2)
the height map on the two-sphere (did not do all the
computations, left as exercise)
R with two atlases: canonical, and {x -> x^3}
identity map not diffeo, but x -> x^3 is (L 2)
an atlas for the two-torus (mainly in pictures)
the open square is one domain, leaves two circles
uncovered
take one more square, and patches around the
two points that are left out
can do something similar for the Klein bottle
maps between (open sets in) vector spaces (L 3)
the differential
higher order derivatives, multilinear maps, the
symmetry
the Taylor Thm in several variables
the "little o" notation
29 M more about the differential and higher order derivatives
for maps in several variables
matrix representation of the differential (the
"Jacobian matrix"; as opposed to the
"Jacobian", which is the determinant of the
Jacobian matrix)
the second derivative, its symmetry, the Hessian
connection with the partial derivatives
the Inverse Function Theorem (statement only)
the "Chain Rule" in several variables
why the Inverse Function Theorem in C^1 implies the one in
C^k
31 W the Implicit Function Theorem ("simple" and "general"
cases)
proofs based on the Inverse Function Theorem
(some of this is done in L 4, but we used a different
notation)
February
5 M more about the above theorems, and the Rank Theorem
(no proof, but example, figures, etc.)
review defn of manifold, differentiable structure,
equivalent atlases, maximal atlas
CONVENTION: unless stated otherwise, all manifolds are
smooth
new charts from old: restriction, pre- and post-composition
new manifolds from old: open subsets, product manifolds
defn of submanifolds (the inclusion becomes an embedding)
[comment: in some books submanifolds are defined so
that the inclusion becomes only an
immersion, as the line of irrational slope
in a 2-torus]
7 W submanifolds, computations for the 2-sphere in R3
critical and singular points, etc.
immersions, submersions, embeddings
examples of the above
12 M more about submersions, immersions and embeddings
(following the typed notes)
lines of irrational slope on the two-torus
14 W the tangent space, in three incarnations:
- as equivalence classes of curves
- as equivalence classes of vectors
- as derivations (for smooth manifolds)
the basis given by a chart
the differential of a map
19 M review & examples of the tangent space, the differential of
a map
the tangent bundle (pages 7-8 of handwritten handout from
last time)
21 W discussion about submanifolds of dimension zero
the tangent bundle
NOTE: error in formula of derivative of Psi on
page 8; moreover, the argument given in the handout
is more complicated then needed
vector bundles and bundle maps (pages 9-13)
did not discuss the compatibility conditions
26 M vector bundles: the compatibility conditions (pages 12-13)
proof of "tangent vectors as derivations", from Naber,
"Tangent vectors and vector fields" section, pages
3-11
definition of vector fields, as (smooth) sections of TM
TO DO FOR NEXT CLASS:
Please read pp 11-20 from Naber. Main points:
the Einstein summation convention (top of page 12)
vector fields as derivations (ex's 81, 82 on p. 16)
the (Lie) bracket of two vector fields (page 17)
its description in local coordinates (ex. on page
17 and 5. on page 19)
Jacobi's identity, 3. on page 19
defn. of a Lie algebra (page 19)
28 W Naber, "Tangent vectors and vector fields"
smooth vector fields (pp. 11-16)
(note: I miss-represented slightly the local
coordinates on TM)
- in local coordinates
- as derivations
the bracket of two vector fields (pp. 17-19)
Lie algebras (pp. 19-20)
TO DO FOR NEXT CLASS:
from CCL $1.4 read pages 29-32, up to Thm. 4.3 (we
discussed this in class today)
from CCL $6.2 read pages 186-192, without the proof
of Thm. 2.3; the important results are
marked with an exclamation mark.
Refresh your memories about ODE's.
Note: the "Derivatives" section of Naber covers
material that we discussed already. You can
find there more details.
March
5 M discussed CCL $6.2, pp 186-192: (local) one-parameter
groups of diffeo's, the relation to vector fields,
the Lie derivative, the bracket of two vector
fields as a Lie derivative
reviewed the needed ODE results
stated the problem solved by the Frobenius Theorem (CCL p 32)
7 W more about ODE's (see pages 130-132, 172-173 of Boothby)
discussed the solutions to HW 2 (on page 4, lines 5 and 6,
replace "row" by "column")
the Frobenius theorem (CCL, pages 32-37):
history: Ferdinand Georg Frobenius (1849-1917),
known also for the Perron-Frobenius
theorem, and many other results (see, e.g.,
Wikipedia)
smooth distributions
the problem, and what the solution means
the necessary condition: the distribution is in
involution
the Theorem: this condition is also sufficient
proof by induction
the case h=1 (Thm. 4.3)
[will do next time: the case h=2]
ERRATA: the definition of the Lie derivative of a vector
field, LXY, that I gave in class on March 5 has
the wrong sign: see CCL page 192, equation (2.22)
for the action of the flow of X on tensors, and
(2.23) for what the Lie derivative is.
Thus, on vector fields one should act with the
inverse of phit, which is phi-t; this changes
the sign of the derivative.
Other news:
The midterm will cover the material up to the
Frobenius theorem. It will be within a few weeks.
For "everything you always wanted to know about
differential geometry", see the five volumes "A
comprehensive introduction to differential
geometry" by Michael Spivak. As of 03/09/2007, a
copy of Volume 1 is on reserve.
19 M handwritten handouts:
partial proof of Thm 4.4 in CCL (Frobenius)
statements and proof of Frobenius in Spivak
- one vector field as d/d x_1
- same for commuting fields
- Frobenius
DISCUSSED:
a few words about vector fields & ODE's in Spivak
Frobenius according to Spivak (sketch of handout)
began alternating tensors from Spivak Ch 7
21 W Spivak Ch. 7, pages 273-281: algebra of alternating forms
Other news: midterm tentatively on Wed., April 4
26 M handwritten handouts:
page about forms
page about exterior differentiatiation
DISCUSSED:
construction of the cotangent bundle (SI, ch 4),
as a special case of the dual of a bundle
exterior differential of a function, df (SI, ch 4)
coordinate free definition (p. 150)
basis in local coordinates for the cotangent space
df in coordinates (Thm. 1, p. 152)
covariant (and more general) tensors, the tensor bundle
(SI, ch 4)
covariant tensors fields as multilinear (over
functions) maps of vector fields to
functions (Thm. 4.2, p. 162)
basis for forms in local coordinates (SI, ch 7, p. 282)
exterior differentiation of forms (SI, ch 7)
in local coordinates (p. 285-286)
why this extends globally (Cor. 12, p. 289)
properties (Prop. 10, p. 286; true globally b/c
definition extends)
coordinate free formula (Thm. 13, p. 289)
(skipped Frobenius thm. stated with forms, p. 292-295)
28 W ERATTA: for a k-form, k is called its degree, not its rank
therefore, change in the handouts from last time
"rk" to "deg"
DISCUSSED:
recalled main points about forms and exterior
differentiation
pull-back and exterior differentiation commute (Prop. 16,
ch. 7, p 295)
closed and exact forms (ch. 7, p. 296-299)
d^2=0 implies that all exact forms are closed
QUESTION: is the converse true?
VAGUE ANSWER:
yes locally (the Poincare Lemma), but not
globally
the "gap" between closed and exact is the
(de Rham) cohomology, which is determined
by the topology of the manifold.
all 1-forms on R^1 are exact (the Fundamental Thm
of Calculus)
in R^2 or R^n: closed 1-forms are exact (p. 298;
also discussed in Calculus III, as
computing the potential of a "curl"-free
vector field)
on (R^2 without the origin) there are closed
1-forms that are not exact (p. 298-299)
the Poincare Lemma:
- a more general statement is Cor. 18, p. 306
(for definitions see p. 300)
- special case: on U an open star-shaped
domain, all closed forms are exact (see
defn on page 300)
proof of the Poincare Lemma for star-shaped
domains:
assume domain is star-shaped wrt zero
define a "degree lowering" map
omega -> theta(omega) given in
exercise 23, p. 321
compute (to do next time):
d(theta(omega)) + theta(d omega) = omega
thus, if d omega=0, then omega is the diff
of theta(omega), and we are done
other news:
midterm on Wed., Apr. 4; closed books
"review" on Mon., Apr. 2; will discuss problems,
bring any questions you have
April
2 M review problems
4 W exam I
9 M handwritten handouts:
6 pages about
d(theta(omega)) + theta(d omega) = omega
2 pages about partitions of unity
DISCUSSED:
solutions to the midterm
d(theta(omega)) + theta(d omega) = omega for k=2, n=3
why we care about the Poincare Lemma:
(de Rham) cohomology, what it is for various
manifolds, and an intuitive explanation why
the Poincare Lemma fails for the plane without the origin
(also discussed on March 28, see references there)
partitions of unity:
defined refinement, locally finite, partition of
unity, subordinate
stated the results (we use for the first time that
a manifold is 2nd countable)
next goal: integration of forms (on an oriented manifold)
and Stokes' formula
orientation for (ordered) bases of a vector space
TO DO FOR NEXT CLASS:
read the handout about orientation from SI Ch. 3
11 W handwritten handouts:
pages about orientation (refers to Ch's 3 and 6 of
SI)
DISCUSSED:
prologue: on an n-manifold one cannot integrate functions
(unless a measure is defined)
however, one can integrate n-forms, if the
manifold is orientable
orientation (results from Ch. 3 and 7 of SI)
why the two-spere is orientable (see it as the
boundary of the unit ball, special case of
orienting the boundary of an oriented
manifold)
details about the orientation of the two-sphere
orientability of the two-torus
the Moebius strip is non-orientable
integration of forms
first, integration of singular k-cubes (defn on p. 334)
motivation (pages 325-333)
the formal definition (page 334)
invariance under "orientation preserving
reparametrizations" of the standard cube
(p. 336)
TO DO FOR NEXT CLASS:
read the integration of chains, up to the Stokes' theorem
(p. 343)
16 M chains and integration on them
the boundary of a chain
18 W Stokes' thm. for chains (no need for orientation)
integration of compactly supported forms on oriented
manifolds
manifolds with boundary
orientation of the boundary
Stokes' thm for oriented manifolds (w/o proof)
23 M integration of compactly supported forms on an oriented
manifold (review)
example of integration; do not have to use only k-cubes
25 W handwritten handouts:
solutions to HW's 4 and 5
DISCUSSED:
proof of the Stokes theorem for manifolds
NOTE: in class we first decomposed the (n-1) form omega in
a finite sum of forms supported inside the n-cubes,
and proved the Stokes formula for each of the terms
(which fall in one of two types). [The computation
can be done on one cube at a time because both
omega and its differential will be supported inside
a cube.] The general case follows by linearity.
In the book, the decomposition is done at the end, in a
different way.
30 M handouts:
typed notes about Sard
Integration on manifolds: theory & practice
solutions to HW 6
DISCUSSED:
the integration on manifolds notes
(the details of integrating forms with compact
support w/o using that the manifold is second
countable - see mention at bottom of page 2 - were
not finished)
the embedding of compact manifolds in R^N (following
Spivak)
the easy Whitney embedding thm: M^n into R^{2n+1}
Last modified: