Many of the exercises are taken from the books referred at the end of the document.

Transcription

1 Exercises in Geometry I University of Bonn, Winter semester 2014/15 Prof. Christian Blohmann Assistant: Néstor León Delgado The collection of exercises here presented corresponds to the exercises for the lecture Geometry I held by Christian Blohmann at the University of Bonn during the Winter Semester 2014/15. Many of the exercises are taken from the books referred at the end of the document. The organization of the exercises in this document is slightly different to the way they were handed during the semester. The former numbering together with the number of points distributed during the tutorials are indicated in brackets. Here the exercises are divided into four topics. Contents 1 Smooth manifolds 2 2 Bundles, vector fields and distributions 4 3 Lie groups and algebras 7 4 Riemannian manifolds and connections 10 1

2 1 Smooth manifolds 1.1. Topological manifolds are paracompact (01-[3 points]) Let M be a connected Hausdorff space that is locally Euclidean, i.e. locally isomorphic to R n. Show that the following are equivalent: (i) M is second countable. (ii) M is paracompact Topological manifolds: a counterexample (02-[3 points]) Show that the set X := {(x, y) R 2 xy = 0} with the subspace topology is not a topological manifold Smooth maps between spheres (03-[3 points]) Show that the following maps are smooth: a) p n : S 1 S 1 the nth power map for n Z, given in complex notation by p n (z) = z n. b) F : S 3 S 2 given by F (w, z) = (zw + wz, iwz izw, zz ww), where we think of S 3 as the subset {(w, z): w 2 + z 2 = 1} of C Complex projective line (04-[3 points]) Show that CP 1 is diffeomorphic to S Homogeneous functions of degree d (06-[3 points]) A smooth function P : R n+1 \ {0} R k+1 \ {0} is said to be homogeneous of degree d (d Z), if P (λx) = λ d P (x) for all λ R \ {0} and x R n+1 \ {0}. Show that the map P : RP n RP k defined by P ([x]) = [P (x)] is well defined and smooth General and special linear group (10-[3 points]) Show that GL(n, R) and SL(n, R) are embedded submanifolds of Mat(n n, R). [Hint: Look at the determinant function.] 1.7. Smooth map with zero differential (05-[3 points]) Let M and N be smooth manifolds, where M is connected. Suppose f : M N is a smooth map. Show that T a f : T a M T f(a) N is the zero map for each a M if and only if f is constant. Give an example showing that the statement is not true if M is not connected The differential in coordinates (07-[3 points]) Let f : M N be a smooth map between smooth manifolds and let a be a point in M. The differential of f at a can be computed in terms of directional derivatives once we have chosen coordinate charts around a and f(a). As seen in the lecture, the differential T a f at a can be computed in local coordinates as map of directional derivatives. Show explicitly that the map does not depend on choice of the charts. 2

3 1.9. Special unitary group (08-[3 points]) Let SU(2) := {A Mat(2 2, C): A A = 1 and det A = 1} denote the group of special unitary 2 2 matrices. (A denotes the Hermitan conjugate of A.) a) Show that SU(2) is a smooth 3-dimensional manifold. b) Show that for a Vector space V there is a natural isomorphism T x V = V for every x V. c) Let i: SU(2) Mat(2 2, C) denote the inclusion. Compute the image of the tangent map T 1 i: T 1 SU(2) T 1 Mat(2 2, C) = Mat(2 2, C). (You can give a basis of the image or linear relations describing the matrices in the image.) Tangent space to a product manifold (09-[3 points]) Let M 1,..., M k be smooth manifolds and let π i : M 1 M k M i, 1 i k denote the projections. Show that for every a = (a 1,..., a k ) M 1 M k the linear map is an isomorphism. T a (M 1 M k ) T a1 M 1 T ak M k v (T a π 1 (v),..., T a π k (v)) Compact manifolds and submersions (12-[3 points]) Let M be a nonempty compact smooth manifold. f : M R n for n 1. Show that there is no submersion 1.12 Embeddings and extensions (E1-[3 points]) Let S be an immersed submanifold of the smooth manifold M. Show that: a) S is embedded if and only if every smooth function on S has a smooth extension to a neighborhood of S in M. b) S is properly embedded (embedded and the map from S to M being proper) if and only if any smooth function on S has a smooth extension to all of M Path lifting property of local diffeomorphisms (11-[3 points]) Let f : M M be a proper local diffeomorphism between non-empty manifolds and x M a point. Show that for every path γ : [0, 1] M through γ(0) = f(x) there is unique path γ : [0, 1] M through γ(0) = x such that the diagram [0, 1] γ γ M f M commutes. (Such a path γ is called a lift of γ.) 3

4 2 Bundles, vector fields and distributions 2.1. The tangent bundle of the 2-torus (13-[3 points]) Show that the tangent bundle of T 2 = S 1 S 1 is trivializable Bundle structure of the Möbius strip (14-[3 points]) As topological space, the Möbius strip M is the quotient of [0, 1] R by the equivalence relation (0, x) (1, x). Show that M has the structure of a smooth vector bundle over S 1 by finding a cover by open trivializations as sketched in the lecture. Determine the structure group of your cover Smooth subbundles (16-[3 points]) Let M be a smooth manifold and S M is an immersed submanifold. Identify T p S as a subspace of T p M for each p S and show that T S is a smooth subbundle of T M S Exterior algebra (17-[3 points]) Let V bea real vector space with dim V = n. Show that the dimension of the exterior algebra (V ) = R V 2 V 3 V... is 2 n Smooth nonvanishing vector field on S 2n 1 (15-[3 points]) For any integer n 1, define a flow on the odd-dimensional sphere S 2n 1 C n by θ(t, z) = e it z. Show that the infinitesimal generator of θ is a smooth nonvanishing vector field on S 2n Vector fields (18-[3 points]) Answer the two following questions: a) Is there a vector field on S 2 vanishing just at a single point? Justify your answer. b) Let a be a smooth manifold M and X X(M) be a vector field on M with flow φ t (m). What is the flow of X? 2.7. Flows in R 2 (19-[3 points]) Compute the flow of each of the following vector fields on R 2 and sketch the flowlines in each case. a) V = y x + y. b) W = x x + 2y y. c) X = x x y y. d) Y = y x + x y. 4

5 2.8. Maximal integral curves (20-[3 points]) Let M be a smooth manifold, X X(M) and γ a maximal integral curve of X. a) We say that γ is periodic if there exists a positive number T such that γ(t+t ) = γ(t) for all t R. Show that exactly one of the following holds: γ is constant. γ is injective. γ is periodic and nonconstant. b) In the later case, show that there exists a positive number T (called the period of γ) such that γ(t ) = γ(t) if and only if t t = kt for some k Z. c) Show that the image of γ is an immersed submanifold of M, diffeomorphic to R, S or R Isotopies: time dependent flows (E2-[3 points]) Let M de a smooth manifold. A smooth isotopy of M is a smooth map H : M J M, where J R is an interval, such that for each t J, the map H t : M M defined by H t (p) = H(p, t) is a diffeomorphism. a) Suppose J R is an open interval and H : M J M is a smooth isotopy. Show that the map V : J M T M defined by V (t, p) = H(p, t) t is a smooth time-dependent vector field on M, whose time-dependent flow is given by ψ(t, t 0, p) = H t Ht 1 0 (p) with domain J J M [Note: the definitions of time-dependent flows and time-dependent vector fields can be found in [LEE] pages ]. b) Conversely, suppose J is an open interval and V : J M M is a smooth timedependent vector field on M whose time-dependent flow is defined on J J M. For any t 0 J, show that the map H : M J M defined by H(t, p) = ψ(t, t 0, p) is a smooth isotopy of M Lie derivative in local coordinates (21-[3 points]) Let M be a smooth manifold. Consider a flow Φ: R M M generated by the smooth vector field v X(M). Let w X(M) be a smooth vector field. Prove by using local coordinates that: L v w := d dt Φ t (w) = [v, w]. t=0 5

6 2.11. Distribution on R 3 (29-[3 points]) Let D be the distribution on R 3 spanned by X = x + yz z, Y = y. a) Find an integral submanifold of D passing through the origin. b) Is D involutive? How does the answer of this question relate to a)? ϕ-invariance (30-[3 points]) Let ϕ: M M be a diffeomorphism. A smooth regular distribution E T M is called ϕ-invariant if T ϕ E : E E is an automorphism of vector bundles. A foliation {S i } i I is called ϕ-invariant if for all i I, ϕ(s i ) = S j for some j I. Show that an involutive distribution is ϕ-invariant if and only if its integrating foliation is ϕ-invariant Integrability of Pfaffian systems (35-[4 points]) a) Show that the kernel of a non-vanishing 1-form α is involutive if and only if α dα = 0. Find an example for a non-vanishing 1-form α Ω 1 (R 3 ) that is not closed, dα 0, but has an integrable kernel. b) Let P = {α 1,..., α k } Ω 1 (M) be a set of 1-forms that are linearly independent at m M. Show that the distribution given by ker(α 1 )... ker(α k ) is involutive if and only if the ideal I Ω(M) generated by P is differentially closed, that is, di I Darboux theorem in 2 dimensions (36-[4 points]) A 2-form ω Ω 2 (M) is called symplectic if it is closed, dω = 0, and non-degenerate, that is, the map T M T M, v i v ω is an isomorphism. a) Show that any non-vanishing 1-form α on a 2-dimensional manifold can be written locally as α = fdg for some functions f and g. Show that this is no longer true if the manifold is of dimension higher than 2. b) Use a) to prove the Darboux theorem in 2 dimensions: Every symplectic form ω on a 2-dimensional manifold can be written locally as ω = dq dp Symplectic and hamiltonian vector fields (37-[4 points]) Let ω Ω 2 (M) be a symplectic form. A vector field v X(M) is called symplectic if L v ω = 0. It is called hamiltonian if there is a function f C (M) such that i v ω = df. a) Show that the Lie bracket of any two symplectic vector fields is hamiltonian. b) Show that if M is non-empty, connected and compact; every hamiltonian vector field vanishes at at least 2 points. 6

7 3 Lie groups and algebras 3.1. Hopf action (22-[3 points]) For any integer n 1 consider the following S 1 action on the odd-dimensional sphere S 2n 1 C n : z (w 1,..., w n ) := (zw 1,..., zw n ). This action is called the Hopf action. Show that it is smooth and its orbits are disjoint unit circles in C n whose union is S 2n SO(2), U(1) and S 1 (23-[3 points]) Prove that SO(2), U(1), and S 1 are all isomorphic as Lie groups Tangent bundle of a Lie group (28-[3 points]) Show that the tangent bundle of every Lie group is trivializable Connected components of a Lie group (31-[3 points]) Let G be a Lie group and G 0 G the connected component containing the unit element. Show that: a) G 0 is a normal embedded Lie subgroup of G. b) There is a bijection between G/G 0 and the set of connected components of G R 3 and so(3) (24-[3 points]) Show that R 3 with the cross product is a Lie algebra which is isomorphic to so(3) SL(2, R) (25-[3 points]) Show that: a) The exponential map is not surjective. exp: sl(2, R) SL(2, R) b) SL(2, R) is connected. [Hint: think of the polar decomposition.] 3.7. SU(2) and SO(3) (26-[4 points]) Show that: a) There is an isomorphism of Lie algebras ϕ: su(2) so(3). b) Construct the homomorphism of Lie groups Φ: SU(2) SO(3) that induces the isomorphism of Lie algebras of a). [Hint: use the exponential map.] c) Show that Φ is a 2-fold covering map. 7

8 3.8. The Heisenberg group (27-[5 points]) Show that: 1 x z a) G := 0 1 y Mat(3 3, R) x, y, z R [G is called the Heisenberg group.] is a Lie subgroup of GL(3, R). b) Compute the Lie algebra g of G. c) Show that the exponential map exp: g G is a diffeomorphism. d) Show that if g is equipped with the product X Y := X + Y [X, Y ] for all X, Y g, exp is an isomorphism of Lie groups Exponential map which is neither injective nor surjective (E3-[3 points]) Show that: cos γ sin γ 0 α a) G := sin γ cos γ 0 β γ Mat(4 4, R) α, β, γ R is a Lie subgroup of GL(4, R). b) Describe the Lie algebra g of G. c) Show that the exponential map exp: g G is neither injective nor surjective Abelian Lie groups and algebras (32-[4 points]) Let G be a connected Lie group and let g be its Lie algebra. a) Show that G is abelian if and only if g is abelian. [Hint: Use that [X, Y ] = 0 if and only if exp tx exp sy = exp sy exp tx for all s, t R for any X, Y g.] b) Show that the center of g is the Lie algebra of the center of G. c) Give a counterexample to a) or b) when G is not connected Graded derivations (33-[3 points]) a) Let A be a graded algebra. Show that the graded vector space A with the graded commutator [a, b] := ab ( 1) a b ba is a graded Lie algebra. b) Show that Der(A) is a graded Lie subalgebra of End(A) with the graded commutator bracket. 8

9 3.12. Odd points of a manifold (34-[5 points]) a) Show that the algebra A := R[ε]/ ε 2 is a Z 2 -graded, graded commutative algebra with deg(ε) = 1. (It is called the algebra of dual numbers or, in supergeometry, the algebra of functions on the odd point R 0 1.) b) Let M be a smooth manifold. Show that there is a natural bijection from vectors in T M to graded homomorphisms of graded algebras, T M Hom ( C (M), A ), where C (M) is viewed as Z 2 -graded algebra concentrated in degree 0. (This is often expressed by saying that tangent vectors are the odd points of a manifold.) c) Generalize this to A := R[ε]/ ε k, where k 2. Show that this is a Z k -graded algebra. What can Hom ( C (R), A ) now be identified with? 9

10 4 Riemannian manifolds and connections 4.1. Piecewise smooth paths as smooth paths (38-[3 points]) Let M be a riemannian manifold and γ : [a, b] M be a piecewise smooth path on M. Show that there exists a smooth path γ : [a, b] M with the same image as γ and hence with equal lenght Submanifold topologies (46-[3 points]) Let N be an embedded submanifold of the Riemannian manifold M. N receives an induced Riemannian metric which defines a metric and a topology on N. Show that this topology coincides with the topology on N induced from the topology of M Christoffel symbols (45-[3 points]) What is the transformation behavior of the Christoffel symbols under coordinate changes? 4.4 Killing vector fields (47-[3 points]) Let (M, g) be a Riemannian manifold. A smooth vector field v on M is called a Killing vector field for g if the flow Φ of v acts by isometries, Φ t g = g for all t. Show that: a) The set of all Killing vector fields on M is a Lie subalgebra of X(M). b) A smooth vector field v is a Killing vector field if and only if it satisfies the following equation in each smooth local coordinate chart: v k g ij x + g v k k jk x + g v k i ik x = 0. j 4.5. Locally but not globally isometric manifolds (E4-[3 points]) Consider the embedding f : R 2 R 3 given by f(x, y) = (x cos y, x sin y, y). The image of f, H with the induced pull-back metric is called the helicoid. Let C R 3 be the catenoid, namely the surface of revolution generated by the curve γ(t) = (cosh t, t). Show that H and C are locally but not globally isometric The hyperbolic space (39-[3 points]) We equip R n+1 with the inner product x, y := x 0 y 0 + x 1 y x n y n for every x = (x 0,..., x n ), y = (y 0,..., y n ) R n+1. Consider the submanifold H n := {x R n+1 : x, x = 1, x 0 > 0}. Show that the pull-back of, to H n is a riemannian metric. (H n is called the hyperbolic space.) 10

11 4.7. Hyperbolic space as unit disk (40-[3 points]) In the notation of the previous exercise, let s := ( 1, 0,..., 0) R n+1 and let D n := {(0, ξ 1,..., ξ n ) R n ξ < 1} be the unit open n-disk embedded into R n+1. Show that: a) The map ξ : H n D n defined by is a diffeomorphism. ξ(x) := s b) The metric assumes in this chart the form g = 2(x s) x s, x s 2 (1 ξ 2 ) 2 dξi dξ i Geodesics on the hyperbolic space (41-[3 points]) Determine the geodesics of H n in the chart given in the previous exercise. [Hint: the geodesics through 0 are the easiest ones.] 4.9. Exponential map on a Lie group (42-[6 points]) Let G be a Lie group endowed with a bi-invariant Riemannian metric (i.e. L g and R g are isometries for every g G). Denote by i: G G the inverse map (i(g) := g 1 for every g G). a) Show that T g i = (T e R g 1)(T e i)(t g L g 1) for each g G. Conclude that i is an isometry. b) Let v T e G and c v be the geodesic satisfying c v (0) = e and c v (0) = v. Show that if t is sufficiently small then c v ( t) = (c v (t)) 1. Conclude that c v is defined in R and satisfies c v (t + s) = c v (t)c v (s) for all t, s R. [Hint: Any two points in a normal coordinate neighborhood are connected by a unique geodesic in that neighborhood.] c) Show that the geodesics of G are the integral curves of the left-invariant vector fields on G. Conclude that the maps exp and exp e coincide Exponential map on the sphere (43-[3 points]) Determine the exponential map of the sphere S n, for example at the north pole N. Answer the following questions: a) What is the supremum of the radii of balls in T N S n on which exp N is injective? b) Where does exp N have maximal rank? 11

12 4.11. Injectivity radius of a surface of revolution (44-[3 points]) Consider the surface S of revolution obtained by rotating the curve (x, e x, 0) equipped with the Riemannian metric induced from the Euclidian metric on R 3. Show that S is complete and compute its injectivity radius. (The injectivity radius at a point p S is the largest radius for which the exponential map at p is a diffeomorphism. The injectivity radius of S is the infimum of the injectivity radii at all points.) 4.12 Hopf-Rinow theorem (48-[3 points]) Let M be a connected Riemannian manifold. Asuume that any two points p, q M can be joined by a geodesic of shortest length d(p, q). Show that this does not imply that M is geodesically complete Flat connections I (49-[3 points]) Show that a linear connection on a vector bundle is flat if and only its curvature vanishes Flat connections II (50-[3 points]) Show that a linear connection of a vector bundle over a 1-dimensional manifold is always flat Bott connection (51-[6 points]) Let F T M be an integrable distribution on M. Let S be one of its leaves. Let NS S denote the normal bundle of S. (Recall that the normal fiber at s S is defined as N s S = T s M/T s S.) a) Show that every vector field w Γ (S, T M S ) supported on S extends to a vector field ŵ on M. Show that every vector field X Γ (S, T S) extends to a vector field on ˆX on M that is everywhere tangent to the leaves of the foliation. b) For every vector field u on M let n(u) denote the induced section of the normal bundle given by n(u) s := u(s)+t s S. With vector fields as in a) show that n([ ˆX, ŵ]) depends only on X and w but not on their (non-unique) extensions. c) Show that D X n(w) := n([ ˆX, ŵ]) defines a covariant derivative on S. (This is called the Bott connection.) d) Show that the Bott connection constructed in c) is flat. Bibliography. [LEE] John M. Lee, Introduction to Smooth Manifolds. Springer (2nd edition, 2013). [JOS] Jürgen Jost, Riemannian Geometry and Geometric Analysis. edition, 2011). Springer (6th 12