Manifolds

Manifolds The Definition of a Manifold and First Examples In brief, a (real) n-dimensional manifold is a topological space M for which every point x ∈ M has a neighbourhood homeomorphic to Euclidean space Rn . Definition 1. (Coordinate system, Chart, Parameterization) Let M be a topological space and U ⊆ M an open set. Let V ⊆ Rn be open. A homeomorphism φ : U → V, φ(u) = (x1 (u), . . . , xn (u)) is called a coordinate system on U, and the functions x1 , . . . xn the coordinate functions. The pair (U, φ) is called a chart on M. The inverse map φ−1 is a parameterization of U. Definition 2. (Atlas, Transition maps) An atlas on M is a collection of charts {Uα , φα } such that Uα cover M. The homeomorphisms φβ φ−1 α : φα (Uα ∩ Uβ ) → φβ (Uα ∩ Uβ ) are the transition maps or coordinate transformations. Recall that a topological space is second countable if the topology has a countable base, and Hausdorff if distinct points can be separated by neighbourhoods. Definition 3. (Topological manifold, Smooth manifold) A second countable, Hausdorff topological space M is an n-dimensional topological manifold if it admits an atlas {Uα , φα }, φα : Uα → Rn , n ∈ N. It is a smooth manifold if all transition maps are C ∞ diffeomorphisms, that is, all partial derivatives exist and are continuous. Two smooth atlases are equivalent if their union is a smooth atlas. In general, a smooth structure on M may be defined as an equivalence class of smooth atlases, or as a maximal smooth atlas. Definition 4. (Manifold with boundary, Boundary, Interior) We define a n–dimensional manifold with boundary M as above, but now allow the image of each chart to be an open subset of Euclidean space Rn or an open subset of the upper half-space Rn+ := {(x1 , . . . , xn ) | xn ≥ 0}. The preimages of points (x1 , . . . , xn−1 , 0) ∈ R+ n are the boundary ∂M of M, and M − ∂M is the interior of M. 1 Manifolds A manifold with boundary is smooth if the transition maps are smooth. Recall that, given an arbitrary subset X ⊆ Rm , a function f : X → Rn is called smooth if every point in X has some neighbourhood where f can be extended to a smooth function. Definition 5. A function f : M → N is a map of topological manifolds if f is continuous. It is a smooth map of smooth manifolds M, N if for any smooth charts (U, φ) of M and (V, ψ) of N , the function ◦ f ◦ φ−1 : φ(U ∩ f −1 (V)) → ψ(V) is a C ∞ diffeomorphism. Exercise # 1. (Recognizing Manifolds) Which of the following have a manifold structure (possibly with boundary)? (a) Arbitrary subset X ⊆ Rn (d) Hawaiian Earring (b) Arbitrary open subset U ⊆ Rn (e) {(x, sin x1 ) | x ∈ (0, 1]} (f) {(x, sin x1 ) | x ∈ (0, 1]} y x S
{0} × [0, 1] z z z x (c) Graph y y x (g) Solution to x2 + y 2 = z 2 (h) Solution to x2 + z 3 = y 2 z 2 2 (i) Solution to z = x2 + y 2 Manifolds A A B A A B A A (j) Quotient Space ABAB −1 (k) Quotient Space AAAA (m) Disjoint union a [0, 1] N of discrete copies of [0, 1] (n) Disjoint union a L N R [0, 1] continuum of discrete copies of [0, 1] (l) Infinite Coordinate Space (o) Infinite Real Hilbert Space It is a difficult fact that not every topological manifold admits a smooth structure. Moreover, a topological manifold may have multiple nondiffeomorphic smooth structures. For example, there are uncountably many distinct smooth structures on R4 . Exercise # 2. (Atlases on the circle) Define the 1–sphere S 1 to be the unit circle in R2 . Put atlases on S 1 using charts defined by: 1. the angle of rotation from a fixed point, 2. the slope of the secant line from a fixed point, 3. the projections to the x and y axes. In each case, compute the transition functions. Exercise # 3. (Topological vs. Smooth) Give an example of a topological space M and an atlas on M that makes M a topological, but not smooth, manifold. Exercise # 4. (Products of manifolds) Let M be a manifold with atlas (Uα , φα ), and N a manifold with atlas (Vβ , ψβ ), and assume at least one of ∂M and ∂N is empty. Describe a manifold structure on the Cartesian product M × N. Exercise # 5. (Manifold boundaries) Let M be an n–dimensional manifold with nonempty boundary ∂M. Show that ∂M is an (n − 1)–dimensional manifold with empty boundary. Exercise # 6. (Atlases on spheres) Prove that any atlas on S 1 must include at least two charts. Do the same for the 2–sphere S 2 . Exercise # 7. (Centering charts) Given a (topological or smooth) manifold M, and any x ∈ M, show that there is some chart (U, φ) with x ∈ U centered at x in the sense that φ(x) = 0. Show similarly that there is some parameterization ψ of U such that ψ(0) = x. 3 Manifolds Definition 6. (Real and complex projective spaces) The projectivization of a vector space V is the space of 1– dimensional subspaces of V . Real and complex projective spaces are defined: RPn := (Rn+1 − {0})/(x0 , . . . , xn ) ∼ (cx0 , . . . cxn ) for c ∈ R CPn := (Cn+1 − {0})/(x0 , . . . , xn ) ∼ (cx0 , . . . cxn ) for c ∈ C Points in RPn and CPn are often written in homogeneous coordinates, where [x0 : x1 : · · · : xn ] denotes the equivalence class of the point (x0 , x1 , . . . , xn ). Exercise # 8. Let Ui = {[x0 : x1 : · · · : xn ] ∈ RPn | xi 6= 0}. Use the sets Ui to define a manifold structure on RPn . Exercise # 9. Show that RPn can also be defined as the quotient of n–sphere S n by the antipodal map, identifying a point (x0 , . . . , xn ) with (−x0 , . . . , −xn ). Exercise # 10. Show that RPn decomposes as the disjoint union RPn = Rn t RPn−1 , and hence inductively RPn = Rn t Rn−1 t · · · t R1 t point. These subspaces are called the point at infinity, the line at infinity, etc. Exercise # 11. Identify among the following quotient spaces: a cylinder, a Möbius band, a sphere, a torus, real projective space, and a Klein bottle. A A B A B A B A A A A B A B B A A B B A Exercise # 12. (The cylinder as a quotient) Define the cylinder C to be the subset of R3 C = {(cos θ, sin θ, z) | 0 ≤ θ < 2π, 0 ≤ z ≤ 1}. Give a rigorous proof that C is homeomorphic to the unit square R = [0, 1] × [0, 1] in R2 modulo the identification (0, x) ∼ (1, x) for all x ∈ [0, 1]. Exercise # 13. (T 2 ∼ = R2 /Z2 ) Show that the quotient of the plane R2 by the action of Z2 is the torus T 2 = S 1 × S 1 ,
T 2 = R2 /(x, y) ∼ (x, y) + (m, n) for (m, n) ∈ Z2 . Exercise # 14. (Challenge) (Classification of 1–manifolds) Prove that any smooth, connected 1–manifold is diffeomorphic to the circle S 1 or to an interval of R. Exercise # 15. (Challenge) (Topological groups) Show that the following groups have the structure of a manifold. Compute their dimension, find the number of connected components, and determine whether they are compact. 1. Real n × n matrices Mn (R) 4 Manifolds 2. Rigid motions of Euclidean space En (R) 3. m × n matrices of maximal rank 4. General linear group GLn (R) = {A ∈ Mn (R) | det(A) 6= 0} 5. Special linear group SLn (R) = {A ∈ Mn (R) | det(A) = 1} 6. Orthogonal group On (R) = {A ∈ GLn (R) | transpose(A) = A−1 } 7. Special orthogonal group SOn (R) = On (R) ∩ SLn (R) Exercise # 16. (Challenge) (The real Grassmannian) The projective space of a vector space V is a special case of the Grassmanian G(r, V ), the space of r–planes through the origin. Show that, as a set,
G(r, Rn ) ∼ = O(n)/ O(r) × O(n − r) . Argue that this identification gives G(r, Rn ) the structure of a smooth compact manifold, and compute its dimension. Multivariate Calculus Definition 7. (Derivative) Given a function f : Rn → Rm , the derivative of f at x is the linear map dfx : Rn −→ Rm y 7−→ lim t→0 f (x + ty) − f (x) t The directional derivative of f at x in the y direction Write f in coordinates f (x) = (f1 (x), . . . fm (x)). If all first partial derivatives exist and are continuous in a neighbourhood of x, then dfx exists and is given by the Jacobian, the m × n matrix  df ··· dx1 (x) 1    .. . dfm dx1 (x) · · ·  df1 dxn (x) .. .    dfm dxn (x) The chain rule asserts that given smooth maps f, g g◦f U /V f g $ /W ∩ ∩ ∩ Rn Rm R` 5 Manifolds then for each x ∈ U, the following diagram commutes: d(g◦f )x Rn dfx / Rm dgf (x) $ / R` Exercise # 17. Let f and g be functions from Rn → R with derivatives dfx and dgx at x. Give formulas for the derivative of their pointwise sum and product, f + g and f g. Conclude that the space of r–times differentiable functions Rn → R is closed under addition and multiplication. Repeat the exercise for functions Rn → Rm under sum and scalar product. Exercise # 18. Let f be an r–times differentiable function Rn → R. Under what circumstances is the pointwise reciprocal f1 an r–times differentiable function? Theorem 8. (Taylor’s Theorem) Given an n-tuple α = (α1 , . . . , αn ), let |α| = α1 + · · · + αn αn 1 xα = xα 1 · · · xn α! = α1 ! · · · αn ! Dα f = ∂ |α| f n · · · ∂xα n 1 ∂xα 1 and let f : Rn → R be a (r + 1) times continuously differentiable function in a ball B of y. Then for x ∈ B, f has a Taylor expansion r X X Dα f (y) (x − y)α + f (x) = Rβ (x)(x − y)β α! |β|=r+1 |α|=0 where the remainder |β| Rβ (x) = β! Z 1
(1 − t)|β|−1 Dβ f y + t(x − y) dt 0 Tangent Spaces and Derivatives Definition 9. (Tangent Space Tx M, Derivatives) Suppose that M is a smooth m-dimensional submanifold of some Euclidean space RN . (We will see in Theorem 18 that every manifold can be realized this way). Let φ : U → M be a local parameterization around some point x ∈ M with φ(0) = x. We define the tangent space Tx M to be the image of the map dφ0 : Rm → RN . Note that Tx M is an m-dimensional subspace of RN ; its translate x + Tx M is the best flat approximation to φ at x. Given a smooth map of manifolds f : M → N , and local parameterizations φ : U → M, φ(0) = x ∈ M and : V → N , ψ(0) = f (x) ∈ N . Let h be the map h = ψ −1 ◦ f ◦ φ : U → V. We can define the differential of f at x by dfx : Tx M → Tf (x) N dfx = dψ0 ◦ dh0 ◦ dφ−1 0 . 6 Manifolds The spaces Tx (M), Tf (x) (N ), and the differential dfx are independent of choice of local parameterizations φ and ψ. Exercise # 19. (Tangent spaces to products) Given smooth manifolds M and N , show that T(x,y) (M × N ) ∼ = Tx M × Ty N . Exercise # 20. (Tangent spaces to vector spaces) Show that if V is a vector subspace of RN , then for x ∈ V , Tx V = V . Exercise # 21. (Chain rule for manifolds) Prove that if f : X → Y and g : Y → Z are smooth maps of manifolds, d(g ◦ f )x = dgf (x) ◦ dfx . There are alternate, intrinsic definitions of the tangent space Tx M that do not require M to lie in RN . Given M and x ∈ M, we define a curve through x to be a smooth map γ : R → M with γ(0) = x. Choose a chart (U, φ), U 3 x; we call two curves γ1 and γ2 through x equivalent if (φ ◦ γ1 )0 (0) = (φ ◦ γ1 )0 (0). Then we can define Tx M as the equivalence classes of curves through x; it is independent of choice of chart. TxM x υ M γ(t) In this formulation, the differential dfx of a map f : N → M is defined as the map dfx : Tx M −→ Tf (x) N [γ : R → M] 7−→ [f ◦ γ : R → N ] The Constant Rank Theorem and Consequences The Constant Rank Theorem states that the local behaviour of a smooth map f : U → Rn is governed by the rank of the Jacobian. Theorem 10. (Constant Rank Theorem) Let U ⊆ Rm be open, and let f : U → Rn be a smooth map. If the Jacobian of f is of constant rank k in a neighbourhood of p, then there are local coordinate systems g at p and h at f (p) such that hf g −1 (x1 , . . . , xm ) = (y1 , . . . , yk , 0, . . . , 0) In other words, in appropriately chosen coordinates, f is locally the composition of the projection Rm  Rk and the inclusion Rk ,→ Rn . 7 Manifolds The case of the Constant Rank Theorem with n = k = m implies the Inverse Function Theorem: Theorem 11. (Inverse Function Theorem) Suppose that M, N are smooth manifolds, and f : M → N is a smooth map whose derivative dfx at x is an isomorphism. Then f is a local diffeomorphism at x, that is, f takes some neighbourhood of x diffeomorphically onto its image. In this case, f locally has a smooth inverse f −1 , with differential d(f −1 )f (x) = (dfx )−1 at f (x). Local diffeomorphism Exercise # 22. (Local diffeomorphism at each point vs. global diffeomorphism) Show by example that even if a map f : X → Y is a local diffeomorphism at each point in X, it may not be a global diffeomorphism. Exercise # 23. (Local Immersion and Submersion Theorems) Suppose that M is an m-dimensional manifold and N is an n-dimensional manifold, m ≤ n. Let f : M → N be a smooth map such that dfx is injective at some point x. Show that the Constant Rank Theorem implies that, in appropriately chosen local coordinates, f is the natural inclusion Rm ,→ Rn in a neighbourhood of x. Similarly, show that if a map g : N → M has a surjective differential dgy at some point y, then there are local coordinates such that g is the projection Rn → Rm in some neighbourhood of y. Definition 12. (Submersion; Immersion) A map f : M → N of smooth manifolds is a submersion if its derivative dfx is surjective at each x. The map is an immersion if the derivative is injective at every x. The image of an immersion is sometimes called an immersed submanifold, though it may not have a manifold structure. E B (p) Submersion (q) Immersion Definition 13. (Embedding, Submanifold) A function f : M → N of topological manifolds is a topological embedding if it is a homeomorphism onto its image. If M and N are smooth manifolds, then f is a smooth embedding if it is an immersion and a topological embedding. Some references additionally require embeddings to be proper maps, that is, preimages of compact sets must be compact. With either definition, the image of an embedding is a (regular) submanifold; the topology on M coincides with the subspace topology on f (M). 8 Manifolds Exercise # 24. (Image of an immersion) Give an example of an immersion i : M → N such that i(M) is not a submanifold of N . Exercise # 25. (Injective immersion vs. Embedding) Give an example of a map of smooth manifolds f : M → N that is injective and an immersion, but not an embedding. Is an injective local diffeomorphism necessarily a global diffeomorphism? Exercise # 26. (Two definitions of embedding) Give an example of a map of smooth manifolds f : M → N that is an immersion and a homeomorphism onto its image, but not proper. Would you call f an embedding? Exercise # 27. (Proper injective immersions are embeddings) Prove that a proper injective immersion of smooth manifolds f : M → N is an embedding. Conclude that if M is compact then the embeddings f : M → N are precisely the injective immersions. Exercise # 28. (Paths in the torus) In Exercise # 13, we realized the torus as the orbit space q : R2 /Z2 ∼ = T 2. Consider the map f : R → T 2 defined by the image of the graph y = mx f : R −→ R2 −→ R2 /Z2 x 7−→ (x, mx) 7−→ q(x, mx) Under what circumstances is this f an immersion? When is it an embedding? When is its image a submanifold? Exercise # 29. The solution sets to the following polynomials are pictured in Exercise # 1: x2 + y 2 = z 2 , x2 + z 3 = y 2 z 2 , and z = x2 + y 2 . Use these formulas to determine which of the solutions sets are embedded submanifolds of R3 . Critical Values and Sard’s Theorem Definition 14. (Critical values, Regular values) Given a smooth map of manifolds f : M → N , the set of critical values of f is defined as {f (p) ∈ N | rank(dfp ) < dim(N )}. The regular values of f are the complement of the critical values in N , that is, {q ∈ N | rank(dfp ) = dim(N ) at every preimage p ∈ f −1 (q)} ∪ {q ∈ N | q ∈ / f (M)}. Theorem 15. (Sard’s Theorem) The set of critical values of a smooth map of manifolds f : M → N has measure 0 in N , in the sense that its preimage in Euclidean space under any local parameterization has measure 0. Exercise # 30. (Critical points, Regular points) Given a smooth map f : M → N , a point p in M is a critical point if rank(dfp ) < dim(N ), otherwise it is a regular point. Note that regular and critical points lie in the domain of f , while regular and critical values lie in the codomain. Show by example that the set of critical points may have positive measure in M. Exercise # 31. (Preimage Theorem) Let M and N be smooth manifolds, and f : M → N . Suppose q ∈ f (M) ⊆ N is a regular value of f . Use the Constant Rank Theorem show that its preimage f −1 (q) is a submanifold of M, with dimension dim(M)− dim(N ). 9 Manifolds Exercise # 32. (Measure of submanifolds) Prove that submanifolds of positive codimension have measure 0. Exercise # 33. (Dense critical values) Show by example that the critical values of a map f : R → R may be dense. Partitions of Unity An essential tool for developing the theory of real manifolds is the existence of partitions of unity: Definition 16. (Partition of unity) Given a smooth manifold M, a partition of unity on M is a set of smooth functions {ρi }I , ρi : M → [0, 1], so that each m ∈ M has a neighbourhood where only finitely many ρi are P nonzero, and I ρi (m) = 1. 1 0 Theorem 17. (Partitions of unity exist) Given a manifold M and an open cover {Uα }A of M, there exists a partition of unity {ρα }A subordinate to U in the sense that the support of ρα is contained in Uα . Alternatively, there exists some partition of unity {ρi }I so that each ρi has compact support contained in some Uα . Partitions of unity allow us to extend certain structures or operations defined in individual coordinate systems to the whole manifolds. Examples are integration, bundle sections (such as vector fields, Riemannian metrics, . . .). Exercise # 34. (Existence of proper maps) Prove that any manifold M admits a proper map ρ : M → R. Whitney Embedding Theorem Theorem 18. (Whitney Embedding Theorem) For m > 0, any smooth m–dimensional manifold admits a smooth proper embedding into R2m . This dimension 2m is a sharp lower bound: RPm cannot embed in R2m−1 whenever m is a power of 2. Exercise # 35. (Challenge) (Weak Whitney Immersion Theorem) Prove a weaker version of Theorem 18: any m–dimensional manifold admits an injective immersion into R2m+1 . Exercise # 36. (Challenge) (Weak Whitney Embedding Theorem) Use Exercises # 34 and # 35 to prove that every m–manifold properly embeds in R2m+1 . Further Reading Munkres, Topology: A First Course. Spivak, Calculus on Manifolds. Guillemin–Pollack, Differential Topology. Lee, Introduction to Smooth Manifolds. Milnor, Topology from the Differential Viewpoint. Madsen–Tornehave, From Calculus to Cohomology. 10