Morse Theory

Question 1

Discuss brief history of Morse theory

Answer 1

Started in 1930’s by Morse. In 1980’s entirely new way of thinking of field emerged –> Floer theory

Question 2

What is Morse theory about?

Answer 2

P = Reasonable space

f: P –> R a reasonably smooth function

Goal: Relate critical points of f to the topology of P (homology, homotopy etc)
can work both ways topology –> critical points and critical points –> topology

In particular prove Morse Inequalities

|Crit_k(f)| >= dim H_k(P)

Inductive move from min to max. See how topology of sublevels change as we go through critical points

pg 9, 19

Question 3

Discuss setting of Morse theory for f.d. manifolds

Answer 3

P = M = Closed manifold.

f at least C^2 normally just C^inf

Crit(f) = {p | df(p) = 0}

Main question: Produce meaningful lower bound for Crit(f)

pg 10

Question 4

What is very weak lower bound on number of critical points of closed manifold?

Answer 4

2 - max min

pg 10

Question 5

What happens if f:M –> R has exactly 2 critical points? M closed manifold

Answer 5

M homeo to S^2, not nec. diffeo

Use Morse theory

pg 10, pg 63

Question 6

Discuss critical points of height function for closed surface. Lower bound? Draw examples

Answer 6

p in Crit(f) <=> T_pP horizontal <– prove

Almost true that >= 2 + 2g

It is possible to construct a function on surface of g >0 with exactly 3 critical points

But is true for broad class of functions –> Morse functions

pg 11-12

Question 7

Discuss Lie derivatives

Answer 7

See Lee book

Question 8

Define: Hessian and discuss/prove properties

Answer 8

quadratic form/bilinear pairing

14-16

Question 9

Can we define Hessian when p is not critical point?

Example?

Answer 9

No - not well-defined

Can find coordinates so f(x) = x_1 <– Hessian = 0

or more generally f(x) = x_1 + any quadratic <– Hessian = whatever you want

pg 15

Question 10

Consider P^n < R^n+1 and f projection onto some coordinate - x_n+1. Discuss how P is locally graph(f)

Answer 10

pg 17

Question 11

Define: p non-degenerate
Morse index
Morse function
Crit_k(f)

Answer 11

pg 18

Question 12

What happens to {f<c} sublevel sets as we move through an interval where c is regular?

Proof?

Answer 12

Nothing happens – diffeomorphism type does not change

Pf. Use anti-gradient flow, key tool of Morse theory. Fix Riemannian metric on P. Now have well-defined gradient that looks the same as classical gradient as long as we choose nice coordinates

Scale gradient to have speed -1 everywhere. Call this X.

phi^t = flow of X

By construction, f decreases along the flow lines of X with unit speed => f(phi^t(x)) = f(x) - t

phi^(b-a) gives diffeo

pg 20 -21

Question 13

What is Morse Lemma? Importance? Examples?

Answer 13

Answers question of what happens at critical points

Prop. p in Crit(f) is non-degen => exists a coordinate system s.t. f(x) = f(p) + 1/2 (-x1^2 - x2^2 - … - xk^2 + xk+1^2 + … + xn^2) = f(p) + 1/2 d_p^2f

k = ind_p (f)

By local change of coordinates at p, eliminate higher order terms in Taylor expansion

going from d^2_pf to (-x1^2 - x2^2 - … - xk^2 + xk+1^2 + … + xn^2) just linear algebra - diagonalize

Index is only local invariant of non-degen critical point. Two critical points with same index are locally diffeo

Notice in examples critical points f = 0 lie on a variety with singularities – not smooth. All other level sets give smooth manifolds

pg 22-23

Question 14

What is Hadamard Lemma? Why is result non-obvious? Proof?

Corollary?

Answer 14

Lemma. f: R^n –> R, f(0) = 0. Then there exist g_i: R^n –> R s.t. g_i(0) = df/dx_i(0) and f(x) = sum g_i(x) x_i near 0.

Can replace R^n by any star-shaped neighborhood.

Pf. FTC used cleverly

Corollary. Assume f(x) = f(0) + h_ij(x)x_ix_j

Note: In general, Hadamard lemma takes over role of converging Taylor expansion

pg 24-27

Question 15

Prove Morse Lemma

Answer 15

1. The question is local - work in R^n
2. f(x) = Q(x) + R(x). Want a diffeo phi so f o phi = Q
3. Use Moser Homotopy Method
4. Set f_t = Q + tR <– homotopy between Q and f. Look for family of diffeos phi_t s.t. f_t o phi_t = Q
5. Differentiate and look for a generating time dependent vector field v_t
6. Get linear system and solve

pg 28 - 32

Question 16

Discuss normal forms and singularities of functions

Answer 16

Consider examples
1. Matrices - Jordan normal form

2. Symmetric matrices - diagonal with +1, -1, 0 (or eigenvalues if we only allow orthogonal transformations)
3. Functions near 0 (germs) - ?

Now (3) in more detail:

(a) df_p != 0, then can write f(x) = x1 + c

(b) df_p = 0. Morse Lemma provides normal form as long as Hessian non-degenerate - only invariant is index

(c) Can go down hierarchy to more and more degenerate functions - classification of singularities very popular in 1970s - difficult

pg 33-35

Question 17

Discuss attaching cells to topological space. Define CW complex. Discuss cell structures on surfaces and projective spaces over R and C

Answer 17

Just form disjoint union and quotient by relation. Build CW inductively, attaching cells in increasing dimension

pg 36-38

Question 18

Discuss 2 criteria for homotopy equivalence from Hatcher

Answer 18

See chapter 0 of Hatcher

Question 19

Prove: Attaching a cell to a CW complex arbitrarily gives a space homotopic to a CW complex

Answer 19

If we are attaching a k+1 cell, we need to push the attaching map, which might hit cells of dim > k, down into k-skeleton.

Idea: If S^k hits interior of ball, we can homotope S^k to boundary of ball by radially projection from point of ball that is not in image of S^k. Then continue pushing inductively until we have pushed into k-skeleton

Problem: Space filling curves.

Solution: Approximate attaching map by map which is smooth on part mapped into ball (Weierstrass approximation theorem). Then Sard’s thm shows curve not space filling.

pg 39-41

Question 20

Discuss cellular homology intuitively - chain complex, differential

Answer 20

C_k = free module over Z generated by the k-cells

differential = sum degree of attaching maps * e_k-1

pg 42-43

Question 21

Discuss cellular homology with differential coming from l.e.s. of pair. Prove we get a chain complex…

Answer 21

See Chris/Hatcher
Attempt to do from purely geometric description as well…he failed so don’t spend too much time

pg 44-46

Question 22

Show cellular homology = singular homology whenever both are defined

Answer 22

So cellular homology is a homotopy invariant.

See Chris/Hatcher

pg 47-49

Question 23

Use CW structure to calculate homology groups of surfaces and projective spaces over R and C

Answer 23

See Chris/Hatcher

pg 49

Question 24

What can we say about topology of a sublevel containing a critical point? Corollaries? Proof of corollaries? Don’t prove main.

Answer 24

Thm. If only critical value in [a,b] is c, then Pb obtained from Pa by attaching a k-cell for every critical point x on the critical level f = c, where k = index(x).

Cor. P has homotopy type of CW complex with one cell for each crit pt with dim = index

In essense exists a complex (C_k, d) with C_k generated by Crit_k(f) over F

Cor. Morse Inequalities. |Crit_k(f)| >= dim H_k(P, F) = b_k or rk H_k((P, Z).

pg 50-51

Question 25

Discuss refining Morse inequalities

Proof?

Answer 25

Consider 2 polynomials:

h(t) = sum dimH_k(P) t^k
m(t) = sum |Crit_k(f)|t^k

the Poincare polynomial and the Morse polynomial. Call coeffic of h b_k - betti and of m C_k

Prop. (1) There exists a poly r(t) “remainder” with coef >= 0 s.t.

m(t) = (1+t)r(t) + h(t) <– in particular coeff of m(t) >= h(t)

(2) Ck- Ck-1+Ck-2-… -+ C0 >= bk-bk-1+…

Cor. Euler Char of complex = Euler Char of manifold. Plug in -1

pg 51-52

Question 26

Prove passing through a critical point adds a k-cell.

Answer 26

Idea: Use unit speed flow along gradient as we did for case without singularities. Use cut-off functions to ensure small neighborhood of singularity remains fixed.

This unretracted neighborhood can be homotoped to a disc…

pg 53-55

Question 27

Discuss handles - examples of handles - and how Morse theory gives handle body decomp. Examples of handle body decomp for surfaces?

Answer 27

A handle decomposition is to a manifold what a CW-decomposition is to a topological space—in many regards the purpose of a handle decomposition is to have a language analogous to CW-complexes, but adapted to the world of smooth manifolds. Thus an i-handle is the smooth analogue of an i-cell. Handle decompositions of manifolds arise naturally via Morse theory.

The cellular decomp we obtain in Morse theory is actually a handle body decomp - attach a k-handle for each critical pt of index k

Cor. A closed smooth manifold has a “handlebody decomposition”

pg 55-57

Question 28

Why doesn’t Morse lemma matter?

Answer 28

Already know that near a non-degen critical point the smooth and top picture is completely dominated by quadratic term - local dynamics

pg 57

Question 29

Define perfect Morse function. Discuss motivation, steps to make perfect, possible? Examples? h-cobordism?

Answer 29

Start with some Morse function f. Want the simplest possible handle-body decomp - smallest number of singularities.

f is PERFECT if d=0 in the associated complex: |crit_k(f)| = b_k

Example surfaces. In reality perfect functions are rare - normally not possible to modify an in imperfect f to a perfect f.

2 steps always work to simplify:
1. Kill extra max’s and min’s
2.Can move lower index pts below higher index pts

In geometric topology and differential topology, an (n + 1)-dimensional cobordism W between n-dimensional manifolds M and N is an h-cobordism (the h stands for homotopy equivalence) if the inclusion maps
M –> W and N –> W
are homotopy equivalences. (Recall: Two manifolds of the same dimension are cobordant if their disjoint union is the boundary of a compact manifold one dimension higher)

The h-cobordism theorem gives sufficient conditions for an h-cobordism to be trivial, i.e., to be C-isomorphic to the cylinder M × [0, 1]. Here C refers to any of the categories of smooth, piecewise linear, or topological manifolds.

The theorem was first proved by Stephen Smale for which he received the Fields Medal and is a fundamental result in the theory of high-dimensional manifolds. For a start, it almost immediately proves the generalized Poincaré conjecture.

pg 58-59

Question 30

Discuss classification of closed surfaces via Morse theory

Answer 30

Hirsch

pg 60

Question 31

Discuss the Poincare conjecture and Morse theory

Answer 31

Milner

pg 60

Question 32

Discuss Heegard Splitting and Morse theory. Proof?

Answer 32

For every closed orientable 3-mfld P, exists a diffeo phi: S_g –> S_g s.t. P = union of 2 solid bodies with g handles glued via diffeo phi.

Proof. Mores theory…nice

pg 60-62

Question 33

What is Morse complex? Goal? Idea? Do examples of torus and weird sphere

Answer 33

Form a chain complex with chains generated by critical points.

Goal: describe differential explicitly without using algebraic topology or CW complex structure

Idea: m(x,y) = number of trajectories from x to y. <– this depends on choice of metric

If index differs by 1, there are generically just a finite number of anti-gradient trajectories from x to y.

pg 65-66

Question 34

Define: mu(x,y). Key theorem? Proof?

Stable, unstable manifolds?

Answer 34

mu(x,y) = { z | phi_t(z) goes to x as t –> -inf and y as t –> inf}

where phi_t is the anti-gradient flow of f and x, y critical points.

Call mu(x,y) moduli space

Thm. For a generic metric, mu(x,y) is a smooth manifold of dimension mu(x) - mu(y) <–difference in index.

stable = incoming trajectories
unstable = outgoing trajectories

Each of these diffeo to open balls - by Morse Lemma locally just quadratic … think of parabolas on different axises

mu(x,y) = W^u(x) int W^s(y) and with a generic metric, we can ensure these sub-manifolds intersect transversely

=> m(x,y) is a manifold and dim mu(x,y) = n - dim W^u(x) - dim W^s(y) = … = mu(x) - mu(y).

pg 68-70

Question 35

Discuss trajectories of height function on torus. Stable/unstable manifolds

Answer 35

Generically, we shouldn’t have trajectories between critical points on saddle.

Tilt slightly.

pg 70

Question 36

Discuss how to order critical points s.t. lower points have lower index.

Answer 36

Sliding handles. We start with an arbitrary Morse function f and modify it by moving critical points past each other

pg 71-72

Question 37

Discuss the modern perspective on mu(x,y)

Answer 37

think of as set of solutions of antigradient equation instead of thinking of as submanifold of P.

mu(x,y) = space of parameterized trajectories

Note: we have a free R-action on mu(x,y). Taking quotient get space of unparameterized trajectories mu hat(x,y) = mu(x,y)/R

This is a smooth manifold of dim mu(x)-mu(y) - 1.

In particular, when m(x) - mu(y) = 1, this is a 0-dim manifold = discrete collection of pts

pg 73

Question 38

Discuss compactification of mu hat (x,y). Importance?

Answer 38

This is a key result which requires the modern view - m(x,y) not being subset.

Thm. For a generic metric, mu hat has a compactification formed by broken trajectories – form manifold with corners.

In the case where index differs by 1, there are no broken paths. mu hat already compact –> finite set. Only finitely many trajectories from x to y.

pg 74

Question 39

Define Morse differential over Z

Do examples of torus and weird sphere

Answer 39

Need to account for orientations. Fix orientations of stable and unstable manifolds by orienting tangent spaces. This gives orientation of mu(x,y)

Trajectories also oriented by flow so we have 2 orientations on each trajectory. If they agree +1 if they disagree -1.

pg 75-76

Question 40

Show Morse boundary^2 = 0, i.e. the Morse complex is actually a complex

Answer 40

Explicitly computing we see that we need to sum over broken trajectories with one brake between critical points whose index differs by 2. mu hat (x,y) is a 1-dim manifold in this case. Compactification is either S^1 or closed interval. If S^1, no broken trajectories - already comapct. Otherwise, we have a pair of broken trajectories - two end points of interval…

pg 77

Question 41

Show Morse homology and cellular homology are the same

Answer 41

Claims differential is the same <– try to check. Note: we proved cellular = singular = topological invariant. So we see Morse homology is a topological invariant of manifold — actually homotopy invariant.

pg 78

Question 42

Do examples of computations with Morse homology – surfaces, projective spaces Real and Complex

Answer 42

pg 79-81

Question 43

Discuss Kunneth formula. Proof?

Answer 43

Kunneth: P0, P1 closed manifolds. Then H_(P0 x P1) = H_(P0) (x) H_*(P1).

Proof. Choose Morse functions f0, f1 on P0, P1 and generic metrics. Product metric on P0 x P1.

pg 83

Question 44

Discuss Poincare duality. Proof?

Answer 44

P smooth closed manifold - homology in dim k matches cohomology in codim k. Palindromic

b_k = b_n-k

Pf. Idea: Choose a Morse function f and look at f and -f. Gradient flows in opposite direction – critical points flip index.

pg 84

Question 45

What can be said about the existence of Morse functions? (Just statements, no proof)

Answer 45

Thm. If P is a closed manifold, then Morse functions form an open dense subset in C^k(P).

Corollary. Every manifold admits a Morse function. Moreover, every function can be C^k-approximated by Morse functions.

Question 46

Recast the definition of critical points, non-degenerate, and Morse function in terms of graphs and transversality

Answer 46

Identify the 0 section with P

x is critical <=> x in graph(df) int P

x is nondegenerate <=> graph(df) int P transverse at x

f is Morse <=> graph(f) transverse to P

pg 86-87

Question 47

Discuss the idea of proof of Morse function using transversality. Difficulty?

Answer 47

1. Start with f0 and approximate by Morse function
2. Transversality thms => df0 approximated by 1-forms whose graphs intersect P transversely.

Difficulty: Need 1-forms exact i.e. = df

pg 87

Question 48

Define: F transverse to Z in Y

Answer 48

dF and TpZ together span TpY
pg 88

Question 49

Let L0 and L1 in R^3 be two embedded loops. For almost all v in R^3 (L0 - v) int L1 = empty.

Proof?

Answer 49

pg 88-89

Question 50

Discuss transversality theorems - increasing constraints on F (4 settings)

Answer 50

X a closed manifold.

Thm 1. F transverse to Z for open dense set in C^k(X,Y)

Y = X x K > Z, F: X–>K
Thm 2. Graph(f) transverse to Z for an open dense set in C^k(X,K)

K –> Y –> X Fiber bundle, Z < Y, s a section.
Thm 3. s transverse to Z for an open dense set of C^k sections

First jet bundle J^1X = T*X x R –> X
First jet of function j^1f = (df, f)

Thm 4. j^1f transverse to Z for an open and dense set of f in C^k(X) k>= 2

pg 90 - 93

Question 51

Discuss proof of existence, density of Morse functions via Jet transversallity

Answer 51

As a result of Jet transversality, we see that df transverse Z in T*X for an open and dense set f in C^k(X) k>= 2.

Now just take Z= P = zero section and we’re done.

pg 93-94

Question 52

Discuss proof of ordinary transversality thm

Answer 52

Guillemin-Pollock

pg 95-96

Question 53

What is C^k topology?

Answer 53

Look up on internet

Question 54

Discuss how –

Thm: Morse Functions form an open and dense set in C^k(P)

– follows from thm about height functions

Answer 54

Thm. For an open dense (full measure) subset of S^m-1, h_v is Morse.

h_v is projection onto line spanned by v. This can be interpreted as a height function

pg 97-98

Question 55

Prove:

Thm. For an open dense (full measure) subset of S^m-1, h_v is Morse.

In the case P a closed hypersurface

Answer 55

Prop. H-v Morse <=> +-v are regular values of Gauss map G

This implies thm, sense regular values form open dense subset of full measure by Sard’s thm.

Notice Crit(h_v) = points where tangent plane is horizontal = points where unit outer normal is vertical –> v or -v = G^-1(v) U G^-1(-v)

pg 99-102

Question 56

Discuss Sard’s Thm, Normal bundle, unit normal bundle, tubular neighborhoods, Gauss map

Answer 56

Lee Ch 6

pg 103

Question 57

Discuss the proof of the general case

Thm. For an open dense (full measure) subset of S^m-1, h_v is Morse.

Answer 57

With the generalized version of Gauss map, have: h_v Morse <=> +- v regular values of G

as before.

PG 104 - 106

Question 58

Define/discuss cup product and intersection product

Answer 58

One way to think of cup product is wedge product of differential forms.

If closed manifold, have Poincare duality - can use to define product in homology

Want to think of intersection product as intersection of 2 chains yielding another chain –> think of as everything as submanifolds

Also read in Hatcher

pg 107

Question 59

Discuss 2 interpretations of intersection product in terms of Morse homology

Answer 59

This requires modern perspective in terms of Morse complex, not just handlebody stuff.

1. Think of Morse homology groups as module over geometric homology groups. - product obtained by looking at trajectories from x to z that pass through B…
2. Use 3 Morse functions to define product…

pg 107 - 111

Question 60

What is the setting for Lusternik-Schrirelmann theory? Compare to Morse theory.

Answer 60

Again we have f:P–>R and want to understand lower bound on |Crit(f)| but here we do not require f non-degenerate.

Underlying idea LS and Morse the same: change in topology from sublevel Pa to Pb => critical pts between a and b.

LS better adapted to inf dimensional settings – think variational problems

Morse deeper connection with topology in finite dims

pg 112

Question 61

Compare lower bounds on critical pts for surfaces, spheres, projective, tori for Morse and LS

Answer 61

In general, LS very far from sharp but any P^n admits f with |Crit(f)| <= n+1. By this, the examples given are sharp.

pg 113

Question 62

Define: LS category, examples? Pros/cons

Answer 62

minimum size of an open cover by contractible sets (IN AMBIENT SPACE - S^1 can be contractible) , Contractible sets need not be connected - may have nontrivial topology

cat(S^n) = 2
cat(surface) = 3
cat(CW) <= n+1

Difficult to compute - upper bounds by explicit constructions. Lower bounds??

Advantage: easy definition

pg 114 - 115

Question 63

Define: cup length, examples?
advantages, disadvantages

Answer 63

max product of cohomology classes of positive degree before getting 0.

Clearly between 0 and dim X

depends on ground field

Easier to calculate. Harder to define - need cohomology

pg 116 -117

Question 64

Discuss with proof the relationship between LS category and cup length

Answer 64

cat(X) >= cl(X) + 1

claim: A < X contractible to a point => H(X,A) –» H(X) onto.

Note: It need not be 1-to-1. Consider collapsing a circle to a point on S^2, creates a wedge of two S^2

pg 117-119

Question 65

Discuss cat_X(A). Advantages over cat(A)?

Answer 65

covering A, but contractible in ambient space X.

With this notion, monotone, continuous, homotopy invariant, subadditive…

pg 119

Question 66

Discuss with proof the relationship of |Crit(f)| and cat(P). Corollaries?

LS approach vs min/max approach

Answer 66

Topological approach (LS approach) to proof. P closed manifold.

Thm. Assume that Crit(f) are isolated. Then |crit values of f| >= cat(P).

Cor. |Crit(f)| > = cat(P) >= cl(P) + 1

Inequalities established for some of our favorite examples are sharp.

Pf. Idea: what is responsible for presence of critical points is change in topology.

Proceed by induction. Since critical points isolated, can cover every critical point on a critical level by little geodesic ball which is contractible. Using anti-gradient flow we see the rest is diffeomorphic to sublevel from previous induction step.

pg 120 - 122

Min/Max Approach

By “LS Inequality” since critical pts isolated, intersecting with non-unit homology class strictly reduces the value of the critical value selector. Writing a maximal intersection = cup length, we get k+1 distinct critical values.

pg 138

Question 67

How few critical points can a function on P have? Pf idea?

Answer 67

n+1 critical pts.

pg 123-125

Question 68

What is the min/max principle? Examples

Answer 68

Says that inf sup f|A is a critical value

min f = given by choosing F = collection of all pts of F

max f = given by choosing F = {P}

pg 126-127

Question 69

Prove: Min/Max principle

Answer 69

pg 128-129

Question 70

3 definitions of CRITICAL VALUE SELECTOR?

Example? Equivalence? Properties?

Answer 70

Let alpha in H_k(P) and define F_alpha = { A : [A] = alpha} = {cycles representing alpha}

Def 1. C_alpha(f) = inf_F_alpha max_A f

Def 2. Assume f is Morse. Then A = linear combo of critical points (in Morse homology) <– if P compact as we assume throughout, there are finitely many critical points so inf becomes min and we only have to calculate on finitely many chains. (just max/min over critical points having non-zero coeff in chain)
If f not Morse, approximate by Morse and take limit

Def 3. c_alpha(f) = inf {a <– regular value : alpha in image(H_(P_a) –> H_(P) }

Properties:
1. Criticality
2. monotonicity
3. C^0 continuity
4. subadditivity

pg 130 - 136

Question 71

Discuss/prove LS inequaility

Answer 71

Assume critical points are isolated and beta != fundamental class, then:
C_alpha int beta (f) < c_alpha(f) KEY: inequality strict.

IDEA: intersecting A with B we get a submanifold of positive codim in A. No reason for it to pass through the isolated max point in A. measure 0 chance

pg 137

Question 72

Discuss/prove Courant-Fischer Thm

Answer 72

Thm, Let Q(x) be a quadratic form, f = Q|S^n, and l1 <= l2 <= … <= ln the eigenvalues of Q. Then lk = inf max f|Lk

Pf. Crit(f) = unit eigenvectors
cv(f) = eigenvalues

This is an immediate consequence of Lagrange multipliers…

pg 141-144

Question 73

Discuss the number of distinct closed geodesics on S^2

Answer 73

Original motivation for LS theory. Any metric on S^2 has >= 3 simple closed geodesics.

Pf. Difficult. Cleaned up by Ballman – required tool of curvature forms to develop.
Outline:
1. Look at space Lambda of smooth embedded loops in S^2
2. Lambda ~ RP^2 ~ Space of Great circles
3. Critical points of the length functional L:RP^2 –> R = simple closed geodesics
4. LS theory => |Crit(L)| >= cl(Lambda) +1 >= 2+1 = 3

pg 145-146

Question 74

What is Polais-Smale condition? Importance?

Answer 74

P = f.d. or Hilbert/Banach manifold

f: P –> R satisfies PS if every sequence {xi} < P s.t. |f(xi)| < C < inf and df(xi) –> 0 contains a convergent subsequence.

IMPORTANT: This condition is very fragile. Easily breaks down.

Without PS, change of topology from Pa to Pb does not imply existence of critical points. Loose connection between topology and critical points.

pg 147 - 148

Question 75

Discuss assumptions accompanying PS which guaranty min value obtained (critical point exists)

Answer 75

1. f bounded from below
2. f satisfies PS
3. The flow for anti-gradient-like vector field is defined for all t >= 0.
   => f has a critical pt, actually min is attained

Pf. Start with x in P, Create sequence using anti-gradient-like flow. PS + bounded from below ==> converges to a critical point y.
pg 149

Question 76

Discuss min/max principle in PS setting. Proof?

Answer 76

Works out very similarly to min/max in compact case and example of PS

pg 150-152

Question 77

Discuss 4 different definitions of geodesic

Answer 77

Let M be a closed manifold with fixed Riemannian metric.

Def 1. A geodesic is a curve locally minimizing length

Def 2. A geodesic is a critical point of the length functional on space of all curves between p and q.

Def 3. A geodesic is a critical point of the energy functional.

Def 4. 0 acceleration curves

3 => 1,2
pg 153-155, 156, 158

Question 78

How can you think of critical points in inf dim spaces?

Answer 78

The change in function over any variation = 0.

pg 155-156

Question 79

How does a critical point of energy functional come with a natural parameterization?

Answer 79

pg 157

Question 80

Define: injectivity radius

Answer 80

an r > 0 s.t. any pts in a r-ball are connected by a unique geodesic.

radius where exponential map is injective

pg 158

Question 81

What is Cartan’s Thm on closed geodesics? Pf?

Discuss pf via finite approximation and infinite dim approach

Answer 81

Thm. For any non-trivial free homotopy class [alpha] in [S^1, M] there is a closed geodesic representative of [alpha].

Idea: take shortest loop in [alpha] or minimize energy over all geodesics

Pf.

1. Introduce idea of geodesic polygons with edge lengths < injectivity radius
2. Every loop in [alpha] can be approximated by geodesic polygons in [alpha]. Every connected component of loop space contains geodesic polygon ==> P_k int Connected component of alpha = union of components of P_k = smooth manifold with boundary
3. Replace E by cursive E (write CE). See both have same critical points. (broken geodesic actually a geodesic if critical point of CE)
4. Find minimum

pg 159-168

175-179

Question 82

What is Lusternik-Fet thm? Proof?

Discuss pf via finite approximation and infinite dim approach

Answer 82

Thm. If pi_1 = 1, there exists a non-constant closed geodesic.

Pf.
1. Topology of loop space: e: Gamma –> M. Use LES of fibration…
2. State exactly how P_k approximates Lambda –> can get isomorphisms of homotopy groups for all pi_n with n < N <– arbitrary, by taking k big enough.

pg 169 - 179

Question 83

Discuss and prove Bott periodicity of U(n)

Answer 83

pg 202