Chapter 1 - Smooth Manifolds Flashcards
Define: topological manifold
Say M is a TOPOLOGICAL N-MANIFOLD if it has the following properties
1. M is a HAUSDORFF space: for every pair of distinct points in M, there are disjoint open subsets separating them
- M is SECOND-COUNTABLE: there exists a countable basis for the topology of M
- M is LOCALLY EUCLIDEAN of dimension n: each point of M has a neighborhood that is homeomorphic to an open subset of R^n (we can force homeomorphism with open ball or R^n)
Define: coordinate chart, centered at p, coordinate domain, coordinate ball, coordinate map, local coordinates
smooth?
A COORDINATE CHART on M is a pair (U, phi) where U is an open subset of M and phi: U –> U hat is a homeomorphism to an open neighborhood of R^n
CENTERED AT P if phi(p) = 0
COORDINATE DOMAIN the set U. Also call COORDINATE NEIGHBORHOOD
COORDINATE BALL if phi(U) is an open ball in R^n
COORDINATE MAP is phi
The component functions of phi (x1, … , xn) are called LOCAL COORDINATES
We add smooth in front of each of these in case the chart is contained in a maximal smooth atlas
Basic examples of smooth manifolds?
- 0-dimensional manifolds: Just a countable discrete space. Only neighborhood of p in M homeomorphic to an open subset of R^0 is {p} itself, exactly one coordinate map and smooth structure
- R^n: Smooth n-manifold with smooth structure determined by the atlas consisting of the single chart ( R^n, Id) - standard smooth structure
- Graphs of smooth functions: Gamma(f) = { (x,y) in R^n x R^k : x in U and y = f(x) } global coordinate chart furnished by projection onto first factor - hence homeomorphic to U itself - GRAPH COORDINATES
- Spheres: S^n easily Hausdorff and second-countable being a subset of R^n+1 it inherits these properties. Charts: Let U_i^+ = {(x1, …, xn+1) in R^n+1 : xi > 0}, define U_i^- similarly - the set where ith coordinate is negative. Let f: B^n –> R be the continuus function f(u) = sqrt(1 - |u|^2). Then U_i^+ intersect S^n is the graph of function x^i = f(x^1, … , x^i hat , … , x^n+1). Similarly for U_i^- x_i = -f(…)
Thus U_i^+ intersect S^n locally Euclidean and phi_i^+-(: U_i^+- intersect S^n —> B^n given by deleting ith coordinate are graph coordinates for S^n. Yields standard smooth structure on S^n - compute transition maps - Projective Spaces: RP^n is defined as the set of 1-dimensional linear subspaces of R^n+1 with quotient topology determined by the natural map pi: R^n+1 {0} –> RP^n sending each point x to the subspace spanned by x, denoted [x]. Let U_i hat in R^n+1 {0} be the set where x^i != 0 and U_i = pi(U_i hat), this is open since U_i hat a saturated open set and pi a quotient map. Define phi_i : U_i –> R^n by phi_i[x^1, … , x^n+1] = (x^1/x^i, … , x_i hat, … x^n+1/x^i). Well-defined - multiply by constant. Continuous - phi_i after pi cts and characteristic property of quotient. Homeomorphismm - cts inverse pg 6. Now check smooth compatibility of charts
- Product Manifolds: M_1 x … x M_k. If each space is Hausdorff & second-countable, so is finite product. Just need charts. Use product map phi_1 x … x phi_k : U_1 x … x U_k —> R^n_1 + … + n_k
- Tori: special case of (5) T^n = S^1 x … x S^1
- Finite dimensional vector spaces: Any norm on V determines a topology which is independent of the choice of norm. With this topology, V is a topological n-manifold and has a natural smooth structure defined as follows. Each ordered basis (E_1, …. , E_n) for V defines a basis isomorphism E: R^n –> V by E(x) = x^iE_i. This map is a homeomorphism, so (V< E^-1) is a chart….pg 17
- Spaces of Matrices: M(mxn, R) denote the set of mxn matrices with real entries. Identify with R^mn by stringing together all matrix entries in a single row.
- Open Submanifolds: Let M be a smooth n-manfold and U < M be any open subset. Define atlas A_U = {Smooth charts (V, phi) for M s.t. V < U}. Can also just think of intersecting every chart of M with U
- General Linear Group: GL(n,R) is an open submanifold of M(n,R) = R^n^2 - i.e. the set det^-1((-inf, 0) U (0, inf) )
- Matrice of Full Rank: Suppose m < n and let M_m(mxn,R) denote the subset of M(mxn),R) consisting of matrices of rank m. If A in M_m(mxn, R) then A has some nonsingular mxm submatrix. By continuity of det, this same submatrix has a nonzero det on aneighborhood of A in M(mxn,R) which implies M_m(mxn,R) is open in M(mxn,R) so a mn-manifold
- Spaces of Linear maps: Suppose V and W are f.d. vector spaces and let L(V,W) denote the set of linear maps from V to W. Then L(V,W) is itself a f.d. vector space so by (7) is a smooth manifold. Choosing a basis for V and W we can identify with mxn matrices (8)
Define: locally finite, refinement, paracompact
A collection X of subsets of M is stb LOCALLY FINITE if each point of M has a neighborhood that intersects at most finitely many of the sets in X.
Given a cover U of M, another cover V is called a REFINEMENT of U if for each v in V there exists some u in U s.t. v < u.
We say M is PARACOMPACT if every open cover of M admits an open, locally finite refinement.
Key topological properties of manifolds?
KEY PROPERTY
1. Every topological manifold has a countable basis of precompact coordinate balls 7-8
CONNECTEDNESS
- Locally path-connected
- Connected <=> path-connected
- Components = path-components
- Countably many components - each a connected manifold
COMPACTNESS
- Locally compact
- Paracompact
- Fundamental group is countable
Define: transition map, smoothly compatible, atlas, smooth atlas, maximal atlas, smooth structure on M, smooth manifold
Why use maximal atlas?
If (U, phi), (V, psi) are two charts s.t. U intersect V != 0, the composite map psi o phi^-1 is called the TRANSITION MAP from phi to psi.
Two charts are SMOOTHLY COMPATIBLE if either the intersection of their coordinate domains is empty or the transition map between them is a diffeomorphism (in the sense of maps on R^n ie bijection with cts partials of all orders).
An ATLAS for M is a collection of charts whose domain covers M. SMOOTH if any two charts in atlas are smoothly compatible
A smooth atlas on M is MAXIMAL if it is not properly contained in any larger smooth atlas (i.e. any chart that is smoothly compatible with every chart in atlas is already in the atlas)
If M is a topological manifold, a SMOOTH STRUCTURE on M is a maximal smooth atlas. A SMOOTH MANIFOLD is a pair (M, A) where M is a topological manifold and A is a smooth structure on M
Many atlases can yield the “same” smooth structure - i.e. determine the same collection of smooth functions on M. Restore uniqueness by requiring maximal smooth atlas
Note: A smooth structure is an additional piece of data that must be added to a topological manifold. A given topological manifold may have many different smooth structures or no smooth structures
Define: regular coordinate ball
Key topological property of smooth manifolds?
We say a set B < M is a regular coordinate ball if there is a smooth coordinate ball B’ > B closure and a smooth coordinate map phi: B’ –> R^n s.t. for some positive real numbers r < r’,
phi(B) = B_r(0), phi(B closure) = B_r(0) closure, and phi(B’) = B_r’(0)
KEY PROPERTY: Every smooth manifold has a countable basis of regular coordinate balls.
Discuss how to think about coordinate charts on a smooth manifold
Think of as giving a temporary identification between U and U hat. Using this identification, we can think of U simultaneously as an open subset of M and as an open subset of R^n.
Visualize identification by thinking of a grid drawn on U representing the preimages of the coordinate lines under phi.
Represent a point p in U by its coordinates (x1, … , xn) - phi(p) and think of this n-tuple as being the point p
Another way to look at this is that by means of our identification U U hat, we can think of phi as the identity map and suppress it from the notation
Discuss the Einstein Summation Convention
If the same index appears exactly twice in any monomial term, once as an upper index and once as a lower index, that term is understood to be summed over all possible values of that index
Basis vectors: E_i with lower indices
Components of vector: x^i with upper indices
What is Smooth Manifold Chart Lemma? Proof?
Let M be a set, and suppose we are given a collection {U_alpha} of subsets of M together with maps phi_alpha: U_alpha –> R^n such that the following properties are satisfied:
1. For each alpha, phi_alpha is a bijection between U_alpha and an open subset phi_alpha(U_alpha) < R^n
2. For each alpha and beta, the sets phi_alpha(U_a intersect U_b) … are open
3. Whenever U_a int U_b != 0, the map phi_b o phi_a is smooth
4. Countably many of the sets U_a cover M
5. Whenever p, q are distinct points in M, either there exists some U_a containing both p and q or there exists disjoint sets U_a, U_b with p in U_a and q in U_b
Then M has a unique smooth manifold structure s.t. each (U_a, phi_a) is a smooth chart
pg 21-22, 28
Holds for manifolds with boundary as well with obvious modifications
Discuss Grassman Manifolds, do example R^3
Let V be an n-dimensional real vector space. For any integer 0 <= k <= n, we let G_k(V) denote the set of all k-dimensional linear subspaces of V. G_k(V) can naturally be given the structure of a smooth manifold of dimension k(n-k). With this structure it is called GRASSMAN MANIFOLD or simply GRASSMANNIAN. Note G_1(R^n+1) = RP^n.
To construct smooth manifold :
1. Let P,Q be any complementary subspaces of V of dimensions k and n-k. Observe the graph of any linear map X: P –> Q can be identified with a k-dimensional subspace of V which intersects Q trivially. Conversely any k-dimensional subspace intersecting Q trivially is the graph of a unique linear map X: P –>Q.
- Let L(P,Q) denote the vector space of linear maps from P to Q and let U_Q denote the subset of G_k(V) consisting of k-dimensional subspaces whose intersections with Q are trivial. The function assigning a linear map Gamma X:P–>Q to its graph in V is a bijection L(P,Q) U_Q by (1). Setting phi = Gamma^-1 and identifiying L(P,Q) with M(n-k x k, R) = R^k(n-k) we have coordinate charts.
- Now apply Smooth Manifold Chart Lemma
Define: n-dimensional topological manifold with boundary, chart, interior chart, boundary chart, interior point, boundary point, boundary, interior, closed manifold
smooth?
An N-DIMENSIONAL TOPOLOGICAL MANIFOLD WITH BOUNDARY is a second-countable Hausdorff space M in which every point has a neighborhood homeomorphic either to an open subset of R^n or to a (relatively) open subset of H^n
An open subset U of M together with a homeomorphism on to an open subset of R^n or H^n is called a CHART for M. INTERIOR CHART if phi(U) is open subset of R^n, BOUNDARY CHART if phi(U) int boundary(H^n) != 0.
INTERIOR POINT if it is in the domain of some interior chart
BOUNDARY POINT if it is in the domain of a boundary chart that sends p to boundary H^n
BOUNDARY of M is the set of all boundary point
INTERIOR is set of all interior points
A CLOSED MANIFOLD is a compact manifold without boundary and an OPEN MANIFOLD is a noncompact manifold without boundary
We add smooth in front of each of these in case the chart is contained in a maximal smooth atlas
Discuss Topological Invariance of the Boundary vs Smooth Invariance of the Boundary
Topological: If M is a topological manifold with boundary, then each point of M is either a boundary point or an interior point, but not both. Thus boundary M and Int M are disjoint sets whose union is M.
Prove this later.
Smooth: Same statement but proof is easier using inverse function theorem
Discuss how to define smooth structures on manifolds with boundary
Recall that a map from an arbitrary set A in R^n to R^k is said to be smooth if in a neighborhood of each point of A it admits an extension to a smooth map defined on an open subset of R^n.
A smooth structure for M is defined to be a maximal smooth atlas where transition maps are smooth in the sense described above. A topological manifold with boundary with such a structure is called a SMOOTH MANIFOLD WITH BOUNDARY
Discuss line with two origins.
Take lines y = 1 and y = -1. Identify all points except x=0. Then M is locally Euclidean and second-countable, but not Hausdorff. Prove this…
pg 29
