Diff-Geom-F12

Homework is to be submitted every Monday either by email or in class. Randomly selected problems will be graded. Problems will be posted on this webpage along with hints and suggestions. During finals week, there will be a final exam. All proofs must be written up in full detail with correctly applied quantifiers and complete justifications. Proofs by contradiction must clearly state "assume on the contrary" and proofs by induction must clearly state the induction hypothesis as well as the base case. References must be given (including precise page numbers) for any theorems applied in the solving of a problem. Credit must be given to any students that worked provided assistance on a problem. Some of the problems may be completed in class but they should be written up adding all necessary detail and submitted regardless.

On this website, mathematics will be written in LaTeX. $ symbols are placed around all formulas. \phi is just the Greek letter phi. \circ is the follows symbol. U_i is U with a subscript i. R^m is R with a superscript m. \cap is the intersection symbol. \cup is the union symbol. \subset is the subset symbol. \forall is the For all symbol. \exists is the There exists symbol. Brackets { } are used to group parts of formulas together within a subscript or superscript. All mathematical articles are written in LaTeX and so it is worthwhile to become familiar with the notation at the same time that you learn new concepts. Handwritten homework is acceptable and encouraged, however, if you do type your homework, then it must be done in LaTex and submitted as a pdf.

8/27: Lesson I: Smooth manifolds, coordinates [Lee Chapter 1]

Although the textbook defines a manifold using concepts from Topology, one does not need these notions to define a smooth manifold. A smooth manifold is just a metric space with a smooth atlas [This is proven in Chapter 8]. That is a smooth manifold, M, is a separable metric space with a collection of open sets $U_i$ covering $M$ and a collection of homeomorphisms $\phi_i: U_i \to V_i \subset R^m$ such that the transition maps $\phi_i \circ \phi_j^{-1}$ are smooth maps from $phi_j(U_i \cap U_j) \subset R^m$ to $\phi_i(U_i \cap U_j) \subset R^m$. Observe that the metric on M is only used to define continuity here, and for this reason this is a topological definition rather than a metric definition. Students who have studied Topology should use the definition in the book and complete all problems in Chapter 1.

Homework #1: All exercises (1.1-1.8) and problems (1-1 to 1-5) in Chapter 1 are due on 9/5. Students who have not taken at least an undergraduate class in Topology, may skip problems 1.1-1.3 and exercise 1-1 and 1-2, and for exercise 1.4 should prove projective space is a separable metric space and that the charts are continuous. They should also prove that every smooth manifold is Hausdorff ($\forall p, q\in M, \exists$ open sets U about p and V about q such that $U \cap V=\emptyset$). Finally they should prove that every smooth manifold has a countable collection of open sets $U_i$ such that $\forall p, q\in M, \exists$ open sets U_i about p and U_j about q such that $U_i \cap U_j=\emptyset$. This countable collection can be found using the charts, but be sure to prove they are open and describe them in detail.

8/29: Lesson II: Examples [Lee Chapter 1]

9/5: Lesson III: Smooth maps, [Lee Chapter 2]

Homework #2 due 9/10: Exercise 2.1, Exercise 2.3 (outlined in class), Exercise 2.4, Problem 2-1

Those who have topology backgrounds, read about smooth coverings and do Exercise 2.6,

a) Find a diffeomorphism from the open disk $D^n$ to Euclidean space $R^n$ (remember these

spaces have atlases with only one chart that are identity maps so this is clever vector calculus).

b) Show that $F(r,s)=(rcos(s), rsin(s))$ is a diffeomorphism from $(0,1) x S^1$ to $R^2 \ 0$.

c) Find a diffeomorphism from the solid torus $D^2x S^1$ to Euclidean space with a line removed, $E^3\L$ using what we discussed in class: $D^2 x S^1$ diffeomorphic to $R^2 x S^1$ diffeomorphic to

$R x (0,1) x S^1$ diffeomorphic to $R x (R^2\p)$ diffeomorphic to $R^3 \ L$.

9/10: Lesson IV: Lie Groups [Lee Chapter 2]

Although Lie Groups appear to be an abstract concept they form a central role in modern physics: Yang-Mills theory and Quantum Mechanics. They are an important part of the qualifying exam and the homework today consists of problems similar to qualifying exam problems.

Homework #3 due 9/19: Read the Lie Groups section and do

Exer 2.8: Let G be a group that is a smooth manifold such that $f: G x G \to G$ defined by $f(g,h)=g*h^{-1}$ is smooth. Prove G is a Lie Group.

a) Prove that GL(2, R) is a Lie Group

b) Prove that $S^1$ is a Lie group. If we identify points in $S^1$ as pairs $(x,y)$ then

what is the product $(x,y) * (a,b)$?

c) Prove that if G_1 and G_2 are Lie Groups then there is a natural product Lie Group $G_1 \times G_2$

which is a product manifold with an operation defined by $(x,y)*(a,b)=(x*a, y*b)$.

9/12: Lesson V: Partitions of Unity [Lee Chapter 2]

In all the work done before today, we could replace the word "smooth" by $C^k$ (k times differentiable), $C^\infty$ (infinitely differentiable) or "analytic". The partition of unity is a uniquely $C^k$ and $C^\infty$ notion as the bump function on $R$ is $C^\infty$ but not analytic.

Homework #3 continued due 9/19: Read the final section of Chapter 2 but do not worry about Lemma 2.15 unless you have taken topology.

a) On a single coordinate chart, draw a graph of a partition of unity of the real line where the given open sets are (-\infty, -2) (-3,3) (3,5) and (4, \infty).

b) Suppose that $\phi_i: N^n \to [0,1]$ for $i=1,2$ is a partition of unity of $N^n$ with open sets $U_i$,

and suppose $F: M^m \to N^n$ is a smooth map. Prove that $\phi_i \circ F: M^m \to N^n$ is

a partition of unity of $M^m$ with open sets $U'_i=F^{-1}(U_i)$. Here $\circ$ is the follows symbol that looks like a small circle and $F^{-1}$ is the inverse of F in tex notation.

Read the Section on Smooth Coverings and do

c) Prove the three properties of smooth coverings: A any smooth map is a local diffeomorphism and an open map, B: an injective smooth covering map is a diffeomorphism. C: (for those who know topology) A topological covering map is a smooth covering map as long as it is a local diffeomorphism.

9/19: Lesson VI: Tangent Vectors [Lee Chapter 3]

Homework #4 due 9/24: Read everything about tangent vectors and tangent spaces up to the definition of TM. TM, the tangent bundle, will be taught next week.

(1) Prove that if D is a derivation and f is a constant function, then Df=0 (using the defn of derivation on a manifold).

(2) Let \phi: U \subset M \to V \subset R^n be a chart. Let $(x_1,...,x_n)$ be the coordinate functions on R^n.

Let p \in U. Let f: U \to R be smooth. Define the partial derivative in the direction x_i of f to be d/dx_i (f \circ \phi^{-1})

evaluated at \phi(p). Call it \hat{e}_i. Prove this is a derivation at p.

(3) Let C:(-r,r) \to M be a smooth curve passing through p=C(0). Define the derivative of f in the direction of C at 0 to be

d/dt (f \circ C(t)) evaluated at t=0. Prove that this is a derivation at p. We call it C'(0) and call it the tangent vector to C at 0.

(4) Given a chart as in (2), define a curve as in (3) using the chart such that the two derivations are equal (give the same

value for every f).

(5) For real numbers a_1 ... a_n define the derivation ( \sum a_i \hat{e}_i ) f to be \sum a_i (\hat{e}_i f). Prove this is a derivation

at any p \in U. If the a_i are smooth functions on U, then we are defining a smooth vector field on U.

(6) Given a curve $C$ as in (3) and a chart as in (2), find the a_i such that the derivation defined by C agrees with the

derivation defined in (5).

(7) Let M be the standard sphere, and let C(t)=(cos(t)/2, sin(t)/2, \sqrt{3}/2), describe the tangent vector C'(0)

in the coordinate chart: \phi(x_1, x_2, x_3)=(x_1, x_2) and in the coordinate chart \phi_2(y_3, y_1, y_2)=(y_1, y_2).

(8) Let M be the standard sphere, and \phi(x_1, x_2, x_3)=(x_1, x_2) draw \hat{e_1} as tangent vectors to the sphere

by finding the corresponding curves and drawing them as vectors in $R^3$ tangent to $S^2$ (intuitive not rigorous).

(9) Let D be any derivation at p, let a_i= D(x_i \circ \phi). Show D= sum_i a_i \hat{e}_i (see book for a hint). So in

fact the dimension of the space of derivations at a point is the same as the dimension of the manifold and the

$\hat{e}_i$ are basis vectors.

(10) Let D be any derivation at p, find a curve C such that D is the derivation defined by C (use the previous problem)

(11) Let G be a Lie Group. Given a derivation D at e, show that L_g* D is a derivation at g. If D is defined by

a curve $h(t)$ passing through $e$, describe how one defines L_g* D.

(12) Let G be the two dimensional torus, draw a vector V at (0,0) and then draw L_gV at various points.

(13) Let G be a matrix group S0(3), and h(t) a curve such that h(0)=Id matrix, what can one say about the

matrix h'(0)? Describe the linear space of derivations at Id as a linear space of matrices (linear under matrix addition).

This is the Lie Algebra so(3).

(14) Let M be the standard sphere and f(x,y,z)=z. Prove that for any derivation D at the point (0,0,1) we have D(f)=0.

(15) Let M be a manifold and F: M \to R smooth which achieves a minimum value at p. Show that for any derivation

$D$ at p. D(f)=0.

Due on 9/24.

9/24: Lesson VII: Tangent Bundle [Lee Chapter 3]

Homework #5 due on 10/1: Read the rest of Chapter 3.

(1) Let G be a group which is a smooth manifold and suppose that f: G \times G \to G be defined by f(g,h)= gh^{-1}

is a smooth map, prove that G is a Lie Group. (if you did this problem to an A last time, resubmit a photo copy of it)

(2) Prove that $S^1$ is a Lie group. If we identify points in $S^1$ as pairs $(x,y)$ then

what is the product $(x,y) * (a,b)$? (if you did this problem to an A last time, resubmit a photocopy of it)

(3) Show that for $S^1$ the circle that $TS^1$ is a trivial bundle $S^1 \times R$.

(4) Let F: S^2 \to R^3 be the standard embedding. Then F_*: TS^2 \to TR^3=R^6 is well defined.

Using the fact that $F(S^2)=\{(x,y,z): x^2+y^2+z^2=1\} \subset R^3$, describe the image

$F_*(TS^2)={(x,y,z, x', y', z'): FILLIN FORMULAS\} \subset R^6$.

(5) Given G is a Lie Group and v_1,... v_n are a basis of tan vectors at e prove:

(5a) the vectors V_i= L_{g*} v_i are a basis of tan vectors at g.

(5b) the vector fields V_i are smooth vector fields.

(5c) TG=G \times \R^m (a trivial bundle) where $m$ is the dimension of $G$.

(6) Given a vector field V: M \to TM and a diffeomorphism, F: M \to M, we say the field is invariant under the

diffeomorphism if $f_*V=V$. If $M=G$ is a Lie group and $F(h)=L_g(h)=g*h$, we say a vector field which

is invariant under this diffeomorphism is left invariant. Show that a vector field $V$ is left invariant iff

$V = \sum a_i V_i where a_i are constant real numbers and V_i are defined as in 4a.

(7) Let F: M \to R and C:(-r,r)\to M be smooth maps. Let X: U \subset M \to V \subset R^m be a chart with coordinates

(x_1,...x_m) and let Y be another chart with coordinates (y_1,...y_m). Show that the vector V\in T_pM

defined by \sum dF/dx_i d/dx_i is not equal to \sum dF/dy_i d/dy_i but the vector defined by

\sum d(x_i \circ C)/dt d/dx_i is equal to \sum d(y_i \circ C)/dt d/dy_i. This happens because TM was defined

using curves like C and not functions like M. So we do not have a notion of gradient of F sitting in TM. We

will have two ways to resolve this. One is with a cotangent bundle and the other is with a Riemannian tensor.

10/1: Lesson VIII: Overview of Differential Geometry [do Carmo Chapter 0]

Read DoCarmo Chapter 0, which summarizes everything we've done this semester and also introduces the notion of orientation (Defn 4.4)

and the brackets of vector fields. You may wish to start going through the problems 7,8, and 9 at the end of this chapter which will be due October .

Homework #6 due on 10/10: Resubmit graded problem sets with additional work attached with corrections.

10/3: Lesson IX: The Cotangent Bundle [Lee Chapter 4]

Read Lee Chapter 4 defining the Cotangent bundle, metric differentials and pull backs.

Homework due on 10/10: Resubmit graded problem sets with additional work attached with corrections.

If you wish you may start next week's homework problems 1-7 due October 15 since we will have covered this material in class already.

10/10: Lesson X: Line Integrals and Conservative CoVector fields [Lee Chapter 4]

Review do Carmo Chapter 0 Sec 4 and Lee Chapter 4.

Homework #7 are due Oct 15:

(1) Do Problem 4-1 of Lee.

(2) Prove the properties of the differential (Prop 4.8 of Lee)

(3) Prove that the cotangent bundle of an orientable manifold is orientable.

(4) Do problem 7 from doCarmo Ch 0.

(5) Do problem 8 from doCarmo Ch 0.

(6) Do problem 9 from doCarmo Ch 0.

(7) Do problem 4-6 of Lee

(8) Do problem 4-9 of Lee