Chapter 2 - Smooth Maps Flashcards
Define: smooth function f: M –> R^k, C^inf(M), coordinate representation of f
smooth map F: M –> N, coordinate representation of F
What modifications need to be made for manifolds with boundary?
We say that f is a SMOOTH FUNCTION if for every p in M, there exists a smooth chart (U, phi) for M whose domain contains p and such that the composite function f o phi^-1 is smooth on the open subset U hat = phi(U) in R^n
C^inf(M) is the set of all smooth real valued functions on M. Under pointwise addition and multiplication this is a commutative algebra over R
Given a function f: M –> R^k and a chart (U, phi) for M, the function f hat : phi(U) –> R^k defined by f hat(x) = f o phi^-1(x) is called the COORDINATE REPRESENTATION of f. By definition, f is smooth <=> its coordinate representation is smooth in some chart around each point
Now let M, N be smooth manifolds and F:M –> N be any map. We say F is a SMOOTH MAP if for every p in M, there exist smooth charts (U, phi), containing p and (v, psi) containing F(p) s.t. F(U) < V and the composite map psi o F o phi^-1 is smooth from phi(U) to psi(V) - we call this map the COORDINATE REPRESENTATION of F.
For a manifold with boundary, we just use the usual fact that a map whose domain is a subset of H^n is smooth if it admits an extension to a smooth map in a neighborhood of each point (the neighborhood being taken in R^n)
Let U be an open submanifold of R^n. Show that f:U–>R^k is smooth <=> smooth in calculus sense
The key is that the smooth structure on U is generated by a single chart with identity map.
Show that if F:M–>N is smooth, then the coordinate rep of F is smooth for every smooth chart
The definition of smooth guarantees on chart containing p in M has smooth coordinate rep. Intersect with desired chart…
pg 35
Prove: Smooth => Continuous
Around any point, we can reduce to consideration of a coordinate representation. Here we really on basic multivariable calculus to see that smooth => continuous (same trick as single variable essentially)… pg 34
Prove: The following are smooth
- Constant maps
- Identity maps
- Inclusions of open submanifolds
- Compositions of smooth maps
pg 36
Examples of smooth maps?
- Any map from a zero dimensional manifold into a smooth manifold
- the map epsilon: R –> S^1 defined by epsilon(t) = e^2pi*it. w.r.t angle coordinates it is linear
- epsilon^n:R^n –>T^n epsilon(x1, … , xn) = (e^2piix1, … , e^2piixn)
- Inclusion of S^n into R^n+1
- Quotient used to define RP^n
- S^n –> RP^n smooth = composition of 4 and 5
- Projection for a product manifold
Define: diffeomorphism, give example, discuss properties
If M and N are smooth manifolds with or without boundary, a DIFFEOMORPHISM from M to N is a smooth bijective map F:M–>N that has a smooth inverse. We say M and N are DIFFEOMORPHIC.
Example: F: B^n –> R^n
F(x) = x/sqrt(1 - |x|^2)
G(y) = y/sqrt(1+|y|^2)
Coordinate charts are diffeomorphisms
Properties:
- Composition of diffeos is a diffeo
- finite product of diffeos is a diffeo
- Every diffeo is homeomorphism and open map
- Restriction of diffeo to open submanifold is diffeo on to image
- Diffeomorphic defines an equivalence relation on the class of smooth manifolds
- Diffeomorphism Invariance of Dimension: A nonempty smooth manifold of dimension m cannot be diffeomorphic to an n-dimensional smooth manifold unless m = n
- Diffeomorphism invariance of Boundary: Diffeomorphisms send boundary to boundary and restric to diffeomorphism on interior.
pg 38-39
Problem with using smooth glueing lemma? Solution?
For continuous maps, we can glue together maps defined on open or closed subsets as long as they agree on intersection. For smooth maps, we cannot glue together closed sets and obtain a smooth result in general f(x) = x, g(x) = -x.
We need to construct maps that agree on open sets (these are large sets) - which is too restrictive
Partitions of Unity are a solution. They are a tool for blending together local smooth objects into global ones
Discuss cutoff functions and bump functions
Both of these constructions rely on the function f(t) = e^-1/t for t >0 and = 0 for t <= 0. Which is smooth.
Given real numbers r1 and r2 such that r1 < r2, there exists a smooth function h: R –> R such that h(t) = 1 for t <= r1, 0 < h(t) <1 for r1 < t < r2, and h(t) = 0 for t >= r2. Such a function is called a CUTOFF FUNCTION
Given any positive real numbers r1 < r2, there is a smooth function H:R^n –> R s.t. H = 1 on B_r1(0) closure, 0 < H(x) < 1 for all x in B_r2(0) - B_r1(0) closure, and H = 0 on R^n - B_r2(0). Such a function is an example of a SMOOTH BUMP FUNCTION pg42
Define: support of f, supported in U, compactly supported, partition of unity, smooth?
If f is any real-valued or vector-valued function on a topological space M, the SUPPORT of f, denoted supp f, is the closure of the set of points where f is nonzero. If supp f is contained in some set U < M, we say that f is SUPPORTED in U. A function f is COMPACTLY SUPPORTED if supp f is a compact set.
Suppose M is a topological space and let X = (X_a) be an arbitrary open cover of M, indexed by A. A PARTITION OF UNITY SUBORDINATE TO X is an indexed family (psi_a) of continuous functions psi_a: M –> R with the following properties:
(i) 0 <= psi_a(x) <= 1 for all a in A and all x in M
(ii) supp psi_a < X_a for each a in A
(iii) The family of supports (supp psi_a) is locally finite
(iv) sum_a psi_a(x) = 1 for all x in M
Given M a smooth manifold, a SMOOTH PARTITION OF UNITY is one for which each of the functions psi_a is smooth
Define: Bump function for A supported in U for an arbitrary closed subset A < M and open set U < M
Prove existence of smooth bump function for a smooth manifold
If M is a topological space, A < M is a closed subset, and U < M is an open subset containing A, a continuous function psi : M –> R is called a BUMP FUNCTION FOR A SUPPORTED IN U if 0 <= psi <= 1 on M, psi = 1 on A, and supp psi < U.
EXISTENCE OF SMOOTH BUMP FUNCTIONS: Let M be a smooth manifold with or without boundary, For any closed subset A < M and any open subset U containing A, there exists a smooth bump function for A supported in U.
Proof. Let U_0 = U and U_1 = M \ A, and let {psi_0, psi_1} be a smooth partition of unity subordinate to the open cover {U_0, U_1}. Notice psi_1 = 0 on A, so psi_0 = 1 on A. Thus the function psi_0 has required properties.
pg44-45
Prove the Extension Lemma for Smooth Functions
Suppose M is a smooth manifold with or without boundary, A < M is a closed subset, and f: A –> R^k is a smooth function. For any open subset U containing A, there exists a smooth function F: M –> R^k s.t. F|A = f and supp F < U.
Proof. For each p in A, choose an extension f_p : W_p –> R^k that agrees with f on W_p int A and where W_p < U. The family of sets {W_p} U M \ A is an open cover. Take partition of unity subordinate to this. Use to blend together all f_p, throwing out M \ A yields desired function
pg 45
Prove Existence of Smooth Exhaustion Functions
Recall, if M is a topological space, an EXHAUSTION FUNCTION FOR M is a continous function f: M –> R with the property that the set f^-1((-inf, c]) (called a SUBLEVEL SET OF F) is compact for each c in R. The name comes from the fact that as n range over the positive integers, the sublevel sets form an exhaustion of M by compact sets.
Every smooth manifold with or without boundary admits a smooth positive exhaustion function.
Pg 46
What can level sets of smooth functions look like?
Let M be a smooth manifold. If K is ANY CLOSED SUBSET of M, there is a smooth nonnegative function f : M –> R such that f^-1(0) = K.
Problem 2.1
pg 48
