Math 868 Final Exam. Part 1. Complete 5 of the following 7 sentences to make a precise definition (5 points each). Y (φ t ) Y lim
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1 SOLUTIONS Dec 13, 218 Math 868 Final Exam In this exam, all manifolds, maps, vector fields, etc. are smooth. Part 1. Complete 5 of the following 7 sentences to make a precise definition (5 points each). 1. A map f : X Y between manifolds is a submersion if DF p : T p X T f(p) Y is surjective for all p X. 2. A map π : V M between manifolds is a (locally trivial) vector bundle of rank k if each p M has a neighborhood U such that a diffeomorphism φ satisfying (i) π 1 φ = π, where π 1 : U R k U is the projection onto the first factor, and (ii) The restriction of φ to each fiber π 1 (x), x U, is linear. 3. Let X and Y be vector fields on M with flows X φ t and Y ψ t. The Lie Derivative of Y in the direction X is the vector field defined by L X Y = d dt (φ (φ t ) Y Y t) Y or lim t= t t 4. A orientation form for a manifold M is or Y (φ t ) Y lim. t t an n-form σ on M (where n = dim M) that vanishes nowhere (i.e. σ(x) x M). 5. Two maps f, g : M N are homotopic if there is a continuous map H : [, 1] M N such that H(, x) = f(x) and H(1, x) = g(x) for all x M. 6. The Poincaré Lemma states that if a domain Ω R n is contractible, then every closed p-form, p >, is exact (or, equivalently, then H p (Ω) = p >.) 7. Let (M, g) be an n-dimensional compact Riemannian manifold with volume form dvol g. The Hodge star operator is the linear map : Ω p M Ωn p M defined by ω ω, where ω is the unique element of Ω n p M such that η ω = η, ω dvolg η Ω p M.
2 Part 2. Do all 4 of the following Short Problems (8 points each). 1. Give a precise definition of a smooth manifold. A smooth manifold M of dimension n is a metrizable (or second countable, Hausdorff) space with a maximal collection of maps ( charts ) {φ α : U α V α α A} indexed by a set A that satisfy: (a) U α M and V α R n are open, (b) {U α } is a cover of M. (c) Each φ α is a homeomorphism. (d) If U α U β, then φ β φ 1 α : φ α (U α U β ) φ β (U α U β ) is a diffeomorphism. Caution: In (c), one can t say φ α is a diffeomorphism because at that point, U α is only a topological space. 2. Consider R 4 with coordinates (w, x, y, z). Let f : R 2 R 4 be defined by f(u, v) = (u 2 v, v, u, v 3 ). Define a 2-form on R 4 by α = w 2 dy dz + y 3 dy dw 2yw dz dw. (a) Compute f α. Noting that dw = d(u 2 v) = 2uvdu + u 2 dv, dx = dv, dy = du, and dz = d(v 3 ) = 3v 2 dv, and that du du = and dv dv =, one calculates f α = (u 2 v) 2 du 3v 2 dv + u 3 du ( 2uv du + u 2 dv) (2u 3 v)(3v 2 dv) (2uvdu + u 2 dv) = (3u 4 v 4 + u 3 ) du dv 12u 4 v 4 dv du = (15u 4 v 4 + u 5 ) du dv. (b) Find a 1-form β on R 4 such that α = dβ. There are many possible answers, including β = w 2 y dz + y4 4 dw and β = w2 y dz y 3 w dy. 3. (a) Suppose that f : M N is a diffeomorphism between manifolds. Prove that at each point p, Df p is an isomorphism of the tangent spaces. By assumption, there is a smooth inverse map g : N M with f g = Id. and g f = Id. By the Composite Function Theorem, D(f g) = Df Dg = D(Id) = Id and similarly D(g f) = Id. Hence Df p : T p M T f(p) N is a (linear) isomorphism of vector spaces. (b) Prove that R k is not diffeomorphic to R n if k n. Fix p R k. If φ : R k R n is a diffeomorphism, by by (a) Dφ p : T p R k T f(p) R n is a vector space isomorphism. Hence dim T p R k = k is equal to dim T f(p) R n = n.
3 4. (a) Define the DeRham cohomology group H p (M). H p (M) = ker d : Ωp M Ωp+1 M im d : Ω p 1 M Ωp M = {closed p-forms} {exact p-forms} (b) Prove that a map f : M N between manifolds induces a map f : H p (N) H p (M) for each p. Let A p (M) Ω p M denote the vector subspace of closed p-froms. For each ω Ap (N), the pullback f ω is closed (since d(f ω) = f dω = ), so defined a class [f ω] in H p (M). Define a map by L(ω) = [f ω]. This map is linear because L : A p (N) H p (M) L(aω + bω ) = [f (aω + bω )] = [af ω + bf ω ] = a[f ω] + b[f ω ]. It is also constant on equivalence classes in A p (N) because L(ω + dη) = [f (ω + dη)] = [f ω] + [f dη] = [f ω] + [df η] = [f ω]. Hence L induces a linear map H p (N) H p (M). Part 3. Do 4 of the remaining 7 longer problems (11 points each). 5. Use the Preimage Theorem (called the Regular Level Set Theorem in Lee) to prove that the graph of any smooth map f : M R between manifolds is a closed embedded submanifold of M R. Define F : M R R by F (x, t) = f(x) t. Then f 1 () is the graph G f = {(x, t) M R t = f(x)}, F is smooth (it is the sum of two smooth functions), and DF p is surjective at each p = (x, t) G f because DF p ( t ) = t. Hence G f is a closed embedded submanifold by the Regular Value Theorem. 6. Let S be the 2-manifold with boundary consisting of the points of the monkey saddle graph {(x, y, z) y 3 3x 2 y z = } whose (x, y) coordinates lie in the ellipsoidal region { ( } y 2 E = (x, y) R 2 x ) (a) Write down a diffeomorphism f : D S, where D is the unit disk in R 2 with coordinates (u, v). Set f(u, v) = (u, 2v, 8v 3 6u 2 v). This is surjective because S is the graph z = y 3 3x 2 y over E, and is injective with inverse f 1 (x, y, z) = (x, y/2). Thus f is a diffeomeorphism (f and f 1 are polynomials, so are smooth). (b) Orient S with the orientation form σ = dx dy, and D with the orientation form du dv. Is your map f positively oriented? f σ = du d(2v) = 2du dv. Since 2 >, f is positively oriented. (c) Compute x dx dy + y 2 dx dz = S xdx dy + y2 dx dz = D u(du 2dv) + (2v)2 (du S d(8v 3 6u 2 v)). Noting that d(8v 3 6u 2 v)) = 24v 2 dv 12uvdu 6u 2 dv and du du =, this reduces
4 to D 2(u + 48v4 12u 2 v 2 ) du dv. Note that u = because u is an odd function. Now switch to D polar coordinates by u = r cos θ, v = r sin θ and du dv = rdrdθ (since f is positively oriented): = r 4( 4 sin 4 θ sin 2 θ cos 2 θ ) rdrdθ = sin 4 θ sin 2 θ cos 2 θ dθ Integrating by parts using (sin 3 θ cos θ) = sin 4 θ + 3 sin 2 θ cos 2 θ, this becomes 4 11 sin 2 θ cos 2 θ dθ = 11(2 sin θ cos θ) 2 dθ = 11 sin 2 2θ dθ = 11π. 7. This problem is about the definition of vector fields as derivations. (a) Complete the definition: A vector field is a linear map X : C (M) C (M) such that X(fg) = Xf g + f Xg for all f, g C (M). Now let {x i } be local coordinates on an open set U M. (b) Use your answer to (a) to show that [, ] = for all i, j. i j (f) = f i. Then [ i, i ]f = j i (f) j j (f) = i i f i f j (c) Show that if X = X i i and Y = Y i i then [X, Y ] = ( X i Y j i is the vector field defined by j =. i Y i Xj i ) j. Write i, as i. Then i,j [Xi i, Y j j ]f is X i i Y j j f +X i Y j i j f Y j j X i i f Y j X i j i f. Omitting the f, changing indices on the thrid term and using (a), this becomes (X i i Y j j Y j i X i ) j + X i Y j [ i, j ] = (X i i Y j j Y j i X i ) j. 8. Let ω be an n-form on a compact n-dimensional manifold M with orientation form σ. (a) Write down a precise definition of the integral ω. For an n-form η compactly supported on an open set U R n, write η = f dx 1 dx n for some function f, and define U ω to be the ordinary integral f. This is well-defined up to sign by the change-of-variables formula from calculus. In general, choose smooth diffeomorphisms φ α : U α V α into M, with U α R n and {V α } covering M. Let {ρ α } be a partition of unity subordinate to {V α }. Define ω to be ±φ α(ρ α ω) M α U α where, for each α, the sign ± are determined by orientation of φ α. M (b) Describe how to determine the sign. For each α, φ ασ is an n-form on a domain in R n, so can be written φ α = g dx 1 dx n, and g vanishes nowhere on U α because φ α is a diffeomorphism. Then the sign is + if g >, and is is g < on U α.
5 9. (a) What is the DeRham cohomology of the circle S 1? (do not prove). H p (S 1 ) = R p = R p = 1 p, 1 Use this and the Mayer-Vietoris sequence to find H (S 2 ), as follows: (b) Draw a picture showing your choice of U and V. Take U to be the northern 3/4 of S 2 and V to be the southern 3/4 (or U = S 2 \ {south pole} and V = S 2 \ {north pole}). (d) Write down the Mayer-Vietoris sequence relating H (U), H (V ) and H (U V ). H p (S 2 ) H p (U) H p (V ) H p (U V ) H p+1 (S 2 ) (c) From (a) and the axioms of cohomology, what are H (U), H (V ) and H (U V )? Note that U and V are contractible, and that U V retracts to the equator. Hence by the Homotopy axiom and the Point axiom { H p (U) = H p (V ) = H p R p =, 1 (point) = otherwise and H (U V ) = H (S 1 ) is an in (a). (d) Use the resulting long exact sequence to find H (S 2 ). S 2 is connected, so H (S 2 ) = R. The Mayer-Vietoris sequence above is, in part, H (S 2 ) H (U) H (V ) H (U V ) H 1 (S 2 ) H 1 (U) H 1 (V ) H 1 (U V ) H 2 (S 2 ). Using (c) this reduces to In particular, R R R R H 1 (S 2 ) R H 2 (S 2 ). (i) R H 2 (S 2 ) is exact = H 1 2(S 2 ) = R. (ii) For p > 2, H p (S 2 ) = H p (S 2 ) =. (iii) Counting dimensions in the exact sequence R R R R H 1 (S 2 ) shows that H 1 (S 2 ) =. Thus H p (S 2 ) = R p = R p = 2 p, 2 1. (a) Complete the definition: M be a smooth n-manifold with boundary if... See the definition in Lee. (b) Now suppose that M is a smooth n-manifold with boundary. Show that M is a smooth (n 1)-manifold without boundary and that the inclusion M M is a smooth embedding. See the definition in Lee.
6 11. The exterior derivative d is a linear map d : Ω p M Ωp+1 M for each p such that (i) d 2 = (ii) For f C (M), the 1-form df is defined by df(x) = Xf for all vector fields X. (iii) d(ω η) = dω η + ( 1) p ω dη ω Ω p M, η Ωq M. Prove that these properties determine d uniquely, as follows: (a) Fill in the blanks above. (b) Show that these properties determine dω for a 1-form ω. Hint: write ω in local coordinates. First suppose that ω has support in one coordinate chart {x i }. For each i, define a function ω i by ω i = ω( ). Then ω = w i i dx i as follows: for any vector field X = X j, j ω(x) = x j ω( ) = ( X j ω j j, while wi dx i) (X) = ω i X j dx i ( ) = ω j j X j. Then by (i) (iii), dω = dω i dx i ω i ddx i = dω i dx i ; in particular, dω is determined by (i) (iii). In general, let {U α } be a coordinate charts that cover M with subordinate partition of unity {ρ α }. Then α ρ α = 1, and hence ω = ω α where ω α = ρ α ω is a 1-form supported on a coordinate chart. Then dω = dω α (by linearity), and each dω α is determined by (i) (iii) as above. (c) Use induction to prove that these properties determine dω for any p-form ω. This is true for p =, 1 by parts (a) and (b). Assume inductively that it is true for p 1. Fix a p-form ω. As in (b), we can use a partition of unity to show that it suffices to assume that ω has support in a chart with coordinates x i. For each i, let η i be the (p 1) form η i = ι i ω Then ω = i η i dx i. Hence by (iii) and (i), dω = dη i dx i +, so dω is well-defined and unique by the induction hypothesis.
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