# An Introduction to 3-Dimensional Contact Topology. Xiao-Song Lin

Save this PDF as:

Size: px
Start display at page:

Download "An Introduction to 3-Dimensional Contact Topology. Xiao-Song Lin"

## Transcription

1 An Introduction to 3-Dimensional Contact Topology Xiao-Song Lin

2 2

3 Contents Preface 5 1 Basics Contact structures Symplectic structures Symplectification and contactification Local and homotopy rigidity of contact structures Almost contact structures and Gromov s theorem More examples Existence of contact structures in dimension Bennequin Introduction The Hopf contact structure Links, braids, and closed braids Transversal links and transversal closed braids Reeb foliation and Markov surfaces Invariants of Legendrian knots Non-standard contact structures on S Legendrian knots Space of contact elements Maslov index Arnold s invariant J Legendrian knots in R Eliashberg Overtwisted contact structures

4 4 CONTENTS 4.2 Tight contact structures

5 Preface In the summer of 2001, I taught a short course on contact geometry as a part of the Fourth Special Mathematics Lectures in Peking University. This short course was meant to be an introduction to the subject, covering mainly the study of 3-dimensional contact geometry before earlier 1990 s. In the past decade, the study of contact geometry entered into a new stage with the advent of Gromov-Witten theory. The so-called Symplectic Field Theory of Eliashberg, Givental, and Hofer relates the studies of contact geometry and symplectic geometry in a much deeper and boarder setting. In the meantime, the study of 3-dimensional contact geometry using geometric and topological methods is also flourishing. It was with these new developments of contact geometry in mind that we hope to offer this book, which is based on my lecture notes of the aforementioned short course, to a wider readership as an introduction to this fascinating subject. Originated in Hamiltonian mechanics and geometric optics, contact geometry caught geometers attention very earlier. In 1953, S.-S. Chern showed that the structure group of a contact manifold M 2n+1 can be reduced to the unitary group U(n) and therefore all of its odd characteristic classes vanish 1. Since all the characteristic classes of a closed, orientable 3-manifold vanish, Chern posed the question of whether such a manifold always admits a contact structure and whehter there are non-isomorphic contact structures on one manifold in One of the milestones in the study of contact geometry is Bennequin s proof of the existence of exotic contact structures (i. e., contact structures not isomorphic to the standard one) on R 3. Bennequin recognized that the induced singular foliation on a surface embedded in a contact 3-manifold plays a crucial role for the classification of contact structures. This role was further explored in the work of Eliashberg, who distinguished two classes of contact structures in dimension 3, overtwisted and tight, and gave a homotopy classification for overtwisted contact structures on 3-manifolds. Eliashberg also proved that on R 3 and S 3, the standard contact structure is the only tight contact structure. For the majority of 3-manifolds, the classification of tight contact structures is unknown. These works of Bennequin and Eliashberg are the main topics of this book. Studying transverse and Legendrian knots in a contact 3-manifold is the approach that Bennequin used to distinguish non-standard contact structures on R 3 and S 3. A deep result in this direction is the so-called Bennequin s inequality, which implies that 1 Thus, all characteristic numbers of M vanish and M is cobordant to zero. 5

6 6 CONTENTS for the standard contact structure on R 3 and S 3, the self-linking number of a Legendrian knot has a topological upper bound. Thus the study of transverse and Legendrian knots is an interesting and rich subject in its own. We shall give an introduction to this subject.

7 Chapter 1 Basics 1.1 Contact structures A contact structure may be regarded as an atlas for a smooth manifold M whose coordinate changes belong to the pseudogroup of local contact transformations. But the modern point of view is to regard a contact structure as a nowhere integrable hyperplane distribution. Definition 1.1. Let M be a smooth manifold of dimension 2n 1. A smooth 1-form α on M is called a contact form if α (dα) n 1 0 everywhere on M. Definition 1.2. A smooth tangent hyperplane field (a smooth distribution of hyperplanes) C on M is called a contact structure if, for every point of M, there is a neighborhood U and a contact form α on U such that C U = ker(α). In the above definition, at a point x U, we consider the 1-form α as a linear functional on T x M. Thus ker(α) is a hyperplane in T x M. Recall that a tangent hyperplane field D is called involutive if for any two vector fields X, Y on M, X, Y D implies [X, Y ] D. Lemma 1.3. Let α be a 1-form and D = ker(α). Then D is involutive iff α dα 0. Thus, following Frobenius Theorem, a contact structure is nowhere integrable. Proof. By definition, D is involutive iff α(x) = 0 and α(y ) = 0 imply α([x, Y ]) = 0. Since α([x, Y ]) = dα(x, Y ) + X(α(Y )) Y (α(x)), we see that D is involutive iff α(x) = 0 and α(y ) = 0 imply dα(x, Y ) = 0. Now pick three vector fields X, Y, Z on M so that X, Y D and Z is not necessarily in D. Then α dα(x, Y, Z) = α(x) dα(y, Z) α(y ) dα(x, Z) + α(z) dα(x, Y ) 0 iff α(x) = 0 and α(y ) = 0 imply dα(x, Y ) = 0. 7

8 8 CHAPTER 1. BASICS By the proof of Lemma 1.3, we see that if α is a contact form, then dα is non-degenerate on the tangent hyperplane ker(α). Definition 1.4. A contact structure C is called co-oriented if there is an oriented line field L such that C L = T M. Lemma 1.5. A contact structure C is defined by a (global) contact form α by C = ker(α) iff it is co-oriented. Proof. Let C = ker(α). There is a line field L such that C L = T M. The restriction of α on L is nowhere zero. So we may give L a direction so that α > 0 in this direction. So L is an oriented line bundle. No assume C L = T M for an oriented line bundle L. This is equivalent to have a vector field X on M so that nowhere X lies in the hyperplane in C. Since the cotangent bundle T M is the dual of the tangent bundle T M, we may find a 1-form α such that C = ker(α) and α is a contact form. It is easy to verify that if α is a contact form and f is a nowhere zero smooth function on M, then fα is also a contact form. These two contact forms, α and fα, determine the same contact structure on M. Furthermore, if two contact forms α and β determine the same contact structure, then β = fα for a nowhere zero smooth function f. Definition 1.6. Integral submanifolds of dimension n 1 of a contact structure on M are called Legendrian submanifolds. A smooth fibration of M, all of whose fibres are Legendrian, is called a Legendrian fibration. Example 1.7. Let M = R 3 with Cartesian coordinates x, y, z. Then α = dz ydx is a contact form. Indeed, α dα = dx dy dz is equal to the standard volume form of R 3. We will call the contact structure C 0 determined by this contact form the standard contact structure on R 3. It is a Legendrian fibration with fibres x, z = constants. Example 1.8. Let M = R 2 S 1 with coordinates x, y, θ (mod 2π). Then α = (cos θ) dx + (sin θ) dy is a contact form (α dα = dx dy dθ). This is the case when we have M as the space of contact elements of R 2. Like the previous example, it is also a Legendrian fibration with fibres x, y = constants. We will discuss such spaces in details later. A contact structure determines an orientation of the (2n 1)-manifold M via the nowhere zero (2n 1)-form α (dα) n 1. If we change the contact form to fα, since (fα) (d(fα)) n 1 = f n α (dα) n 1,

9 1.2. SYMPLECTIC STRUCTURES 9 the derived orientation does not change when n is even. In such a case, if M is oriented, we say that a contact structure is positive if the orientation derived from the contact structure agrees with the given orientation of M. Otherwise, the contact structure is negative. Definition 1.9. Let α be a contact form. The Reeb vector field of α is a vector field R α determined uniquely the following conditions: α(r α ) = 1 and ι Rα (dα) = 0. Recall ι X is the contraction with respect to the vector field X. I.e., ι X (dα)(y ) = dα(x, Y ) for all Y. Since dα is an odd-dimensional anti-symmetric bilinear form, its determinant under a basis is zero. Thus, we may find a non-zero vector R α so that ι Rα (dα) = 0. This vector can not lie in ker(α) since dα is non-degenerate on this tangent hyperplane. Therefore α(r α ) 0 and we may normalize R α so that α(r α ) = 1. The Lie derivative L X formula: with respect to X on forms satisfies the following Cantan L X = ι X d + d ι X. Thus the conditions characterizing the Reeb vector field can be given as α(x α ) = 1 and L Xα α = 0. Let Φ t be the flow of the Reeb vector field X α. Since d dt (Φ t (α)) = Φ t (L X α (α)) = 0, we see that α is preserved under the flow of X α, i.e., Φ t (α) = α. This is Liouville s theorem. We also have L Xα dα = 0 and Φ t (dα) = dα. Thus, the flow of the Reeb vector field preserves the volume form α dα. 1.2 Symplectic structures Symplectic structures and contact structures are closely related. They share the same property of having no local invariants (Darboux Theorem). In some special cases, symplectic structures and contact structures can be translated from one to another by the processes of symplectification and contactification. In recent years, exploration of this transition is proved to be one of the most fruitful approaches in the study of contact geometry. Definition A symplectic form ω on a smooth manifold W of dimension 2n is a closed 2-form which is nondegenerate everywhere. Thus a symplectic form ω satisfies two conditions: (1) dω = 0; and (2) ω(x, Y ) = 0 for all Y implies X = 0 at every point of W.

10 10 CHAPTER 1. BASICS Definition Let H be a smooth function of W equipped with a symplectic form ω. The Hamiltonian vector field X H associated with H is determined uniquely by the condition ι XH (ω) = dh. The condition defining a Hamiltonian vector field X H implies L XH (ω) = ι XH (dω) + d(ι XH (ω)) = 0. So, if we let Φ t be the flow of the Hamiltonian vector field X H, then Φ t (ω) = ω. This is again Liouville s theorem. Theorem Suppose we have two symplectic forms ω 0 and ω 1 on W. Let x 0 W such that ω 0 x0 = ω 1 x0. Then there is a neighborhood U of x 0 and a diffeomorphism Φ : U Φ(U) such that Φ(x 0 ) = x 0 and Φ (ω 1 ) = ω 0. Proof. Consider the closed form ω t = (1 t)ω 0 + tω 1. This is a closed 2-form and nondegenerate at x 0. Therefore, there is a neighborhood U 1 of x 0 such that ω t is nondegenerate on U 1. Since ω 0 ω 1 is a closed 2-form on U 1, by Poincaré s lemma, there is a 1-form α on U 1 such that dα = ω 0 ω 1 and α x0 = 0. Now we define a time-dependent vector field X t satisfying ι Xt (ω t ) = α. Since ω t is nondegenerate on U 1, this condition determines X t uniquely. We have X t (x 0 ) = 0 for all t. by Let Φ t be the diffeomorphism given by the flow of X t at time t, i.e., it is determined Then, we have d dt Φ t(x) = X t (Φ t (x)) and Φ 0 (x) = x for all x U 1. ( ) d d dt Φ t (ω t ) = Φ t dt ω t + L Xt (ω t ) = Φ t (ω 1 ω 0 + ι Xt (dω t ) + d(ι Xt (ω t ))) = Φ t (ω 1 ω 0 + dα) = 0 Therefore, Φ t (ω t ) = Φ 0(ω 0 ) = ω 0. In particular, Φ 1 (ω 1 ) = ω 0 and Φ 1 (x 0 ) = x 0. So this is the desired diffeomorphism Φ defined in a neighborhood of x 0

11 1.3. SYMPLECTIFICATION AND CONTACTIFICATION 11 Corollary (Darboux s Theorem) Let ω be a symplectic form on W. For every point x 0 W, there is a local coordinate system (p 1,..., p n, q 1,..., q n ) in a neighborhood of x 0 such that x 0 = (0, 0,..., 0) and n ω = dp i dq i. i=1 1.3 Symplectification and contactification Let α be a contact form on M. Let W = M R and t R be the coordinate. Define a closed 2-form on W by ω = d(e t α) = e t dt α + e t dα. Since (ω) n = e nt dt α (dα) n 1, we get a symplectic form on W. This is called the symplectification of the contact form α. Consider a function H : W R such that dh = e t dt. Let X α be the Reeb vector field associated with the contact form α. The Hamiltonian vector field of H is denoted by X H. Lemma We have X H = X α. Proof. We have and (e t dt α + e t dα)(x α, / t) = e t dt( / t) α(x α ) = e t, for every vector field Y of M. Thus and X H = X α. (e t dt α + e t dα)(x α, Y ) = e t dα(x α, Y ) = 0 d(e t α)(x α, ) = e t dt( ) = dh( ) For a symplectic form ω on a manifold W of dimension 2n, a vector field X is called Liouville vector field if L X (ω) = ω, or equivalently d(ι X (ω)) = ω. Thus, if we let α = ι X (ω), this gives rise to a contact form on every submanifold M W of dimension 2n 1 transverse to X. Indeed, α (dα) n 1 = (ι X (ω)) (d(ι X (ω))) n 1 = ι X (ω) ω n 1 = 1 n ι X(ω n ),

12 12 CHAPTER 1. BASICS which is nondegenerate on M. This is called the contactification of the symplectic form ω. Consider the case of W = M R and ω = d(e t α) being the symplectification of the contact form α on M. Lemma X = is a Liouville vector field of the symplectic form ω = d(e t α). t Moreover the contactification of this symplectic form on M {t} is e t α. Proof. We check the condition of X being a Liouville vector field: L X (ω) = d(ι X (ω)) = d(ι X (e t dt α + e t dα)) = d(e t α) = ω. This also shows that ι X (ω) = e t α. Definition Vector fields whose local flows preserve a contact structure are called contact vector fields. Let α be a contact form on M, a smooth function K : M R is called a contact Hamiltonian if there is a contact vector field V such that K = ι V (α) = α(v ). Lemma The correspondence K = ι V (α) maps the space of contact vector fields bijectively onto the space of smooth functions. Proof. Given a smooth function K on M with a contact form α, we want to find a contact vector field V such that K = ι V (α). Let R α be the Reeb vector field of α. The following two equations ι X (α) = 0 and ι X (dα) = ι Rα (dk) α dk determines a vector field X uniquely. Notice that the right hand side of the second equation above vanishes on R α. So the second equation is an equation in ker(α). Now let V = KR α + X. We have L V (α) = K L Rα (α) + dk ι Rα (α) + L X (α) = dk + ι X (dα) = ι Rα (dk) α So V is a contact vector field. We have α(v ) = K and K is a contact Hamiltonian, Denote by S = {x M ; K(x) = 0} and assume that it is a smooth submanifold of M. The tangent space of S is ker(dk). If x S such that ker(α) = ker(dk) at x, then by the construction of Lemma 1.17, V (x) = 0 and x is a fixed point of the contact flow Φ t. 1.4 Local and homotopy rigidity of contact structures Just like symplectic structures, contact structures are all locally isomorphic.

13 1.4. LOCAL AND HOMOTOPY RIGIDITY OF CONTACT STRUCTURES 13 Theorem (Darboux s Theorem) Let α be a contact form on M. For every point x 0 M, there is a local coordinate system (z, p 1,..., p n, q 1,..., q n ) in a neighborhood of x 0 such that x 0 = (0, 0,..., 0) and n α = dz q i dp i. i=1 Proof. This is a consequence of Darboux Theorem for symplectic forms (Corollary 1.13). Suppose α is a contact form defined in a neighborhood U of x 0 M, with R α its Reeb vector field. We can form the quotient smooth manifold U/R α such that π : U U/R α is a trivial principal R-bundle with the connection 1-form α (so that its curvature is dα). The push-forward of dα, ω = π (dα), is a symplectic form on U/R α. Thus, there is a local coordinate system (p 1,..., p n 1, q 1,..., q n 1 ) on U/R α, with π(x 0 ) = (0, 0,..., 0) in this coordinate system, such that ω = n 1 i=1 dp i dq i. We use z to denote the coordinate of the fibre through x 0, z(x 0 ) = 0. Starting at x 0, we lift each coordinate axis of U/R α to U as a geodesic with respect to the connection form α. This gives us a coordinate system (with notation abused) (z, p 1,..., p n 1, q 1,..., q n 1 ) in a neighborhood of x 0, which has coordinates all equal to 0. In this coordinate system α = dz n 1 i=1 q i dp i. One of the earlier significant results in contact geometry is the following theorem of Gray. It was proved first along the line of Kodaira-Spencer deformation theory. The modern proof of this theorem follows the so called Moser trick which is much simpler than the original proof. Theorem (Gray) If α t, t [0, 1], is a smooth 1-parameter family of contact forms on a closed smooth manifold M, then there is a diffeomorphism Φ : M M (isotopic to identity) such that Φ (α 1 ) = fα 0 for some function f 0. Proof. We want to construct a time-dependent vector field X t such that its induced diffeomorphism Φ t satisfies Φ t (α t) = λ t α 0 for some positive function λ t on M. Differentiate both sides of the required equality, we get d dt Φ t (α t) = Φ t ( α t + L Xt (α t )) = λ t α 0 = λ t λ t Φ t (α t). Let µ t = d dt log λ t Φ 1 t,

14 14 CHAPTER 1. BASICS then X t has to satisfy Assume further X t ker(α t ), we get Let R αt Φ t ( α t + d(α t (X t )) + ι Xt (dα t )) = Φ t (µ t α t ). α t + ι Xt (dα t ) = µ t α t. be the Reeb vector field of α t, we get α t (R αt ) = µ t. Thus µ t is completely determined and so is X t. Since M is closed, we can get the diffeomorphism Φ t by solving d dt Φ t(x) = X t (Φ t (x)) for all x M, Φ 0 = Id for all t. Eventually, we have ( t λ t (x) = exp 0 ) µ t (Φ t (x)) dt. Corollary Two (co-orientable) contact structures on a closed smooth manifold M that are homotopic in the space of contact structures on M are isotopic. 1.5 Almost contact structures and Gromov s theorem Denote by T M the tangent bundle of M. Let ξ be a co-orientable, co-dimension 1 subbundle of T M with a complex structure J : ξ ξ (J 2 = Id). The pair {ξ, J} is called an almost contact structure of M. If M admits an almost contact structure, then the structure group of M can be reduced to U(n 1). If α be a contact form on M, then ξ = ker(α) is a co-orientable, co-dimension 1 subbundle of T M. Moreover, dα is a symplectic form on ξ. Then there is a complex structure J on ξ, unique up to homotopy, that is compatible with the symplectic structure dα, i.e. and dα(jx, JY ) = dα(x, Y ) dα(x, JX) > 0 if X 0. for all X, Y ξ This pair {ξ, J} is the almost contact structure induced by the contact form α. Theorem (Gromov) If M is open, then two co-oriented contact structures on M are homotopic iff their induced almost contact structures are homotopic. This theorem is a consequence of Gromov s h-principle.

15 1.6. MORE EXAMPLES More examples Example Consider the 3-dimensional Lie group G = SO(3), the rotation group of the Euclidean space R 3. Its Lie algebra g = T 1 G has a basis: We have X = 1 0 0, Y = 0 0 1, Z = [X, Y ] = Z, [Y, Z] = X, [Z, X] = Y. Thus, translating the subspace of g spanned by X, Y, say, over G by left multiplication gives rise to a contact structure on G. Let γ(s) be an arclength parameterized curve in R 3. Let T(s), N(s), B(s) be the Frenet framing of γ(s). Consider ˆγ(s) = (T(s), B(s), N(s)) as a curve in SO(3). Since 0 κ(s) 0 (T (s), N (s), B (s)) = (T(s), N(s), B(s)) κ(s) 0 τ(s), 0 τ(s) 0 where κ(s) and τ(s) are curvature and torsion of γ(s), respectively, we conclude that ˆγ(s) is a Legendrian curve in SO(3). Example (Lutz) On S 1 R 2, we have the coordinate system (θ, r, φ), where θ is the coordinate of S 1 and (r, φ) is the polar coordinates of R 2. Let α = dθ + r 2 dφ. This is a contact form on S 1 R 2. Indeed, with Cartesian coordinates x 2 + y 2 = r 2, arctan y x = φ, we have and α = dθ + xdy ydx α dα = 2dθ dx dy. The roles played by θ and φ can in fact be switched. This is an important fact for the study of contact structures in dimension 3. Let consider 1-forms of the form α = h 1 (r)dθ + h 2 (r)dφ. We have ( ) h1 (r) h α dα = det 2 (r) h 1 (r) h 2 (r) dθ dr dφ.

16 16 CHAPTER 1. BASICS So, as far as the curve r (h 1 (r), h 2 (r)) R 2 has the property that its position and velocity vectors never parallel, α is a contact form. We may now change the contact form α = dθ + r 2 φ by changing the curve (1, r 2 ) to (h 1 (r), h 2 (r)) so that it spirals around (0, 0) several times and still ends at (1, 0). This new contact form is so-called overtwisted. These contact forms are in fact not homotopic if the number of twists are different. On the other hand, we can have the new curve (h 1 (r), h 2 (r)) equal to (1, r 2 ) for r large but equal to ( r 2, 1) for r small. This is the case when the roles played by θ and φ are getting switched and contact surgery becomes possible. 1.7 Existence of contact structures in dimension 3 The existence of a contact structure on every closed, orientable 3-manifold was asked by S.-S. Chern in 1966 [3]. The question was raised because the obvious obstruction for the existence of a contact strucuture vanishes for a closed, orientable 3-manifold. Theorem (Lutz-Martinet) There is a contact structure on every closed orientable 3-manifold. This theorem was first proved by Lutz and Martinet in The proof given here is due to Thurston and Winkelnkemper. It is based on the following theorem of Alexander on a construction which produces all closed orientable 3-manifold. Theorem Every closed, smooth, orientable 3-manifold M is diffeomorphic to F (h) f F D 2, where F is an orientable, compact 2-manifold with one boundary component F, h : F F is a diffeomorphism which restricts to the identity on F ; F (h) denotes the mapping torus of h, i.e. the 3-manifold with boundary obtained from F [0, 1] by identifying (x, 0) with h(x), 1), and f : F (h) (S 1 D 2 ) is a diffeomorphism. Proof of Theorem We may assume that h fixes a collar neighborhood of C of F, and C has coordinates (t, θ) with t = 0 for points on F. We have h(t, θ) = (t, θ). Let α 1 be a 1-form on F equal to (1 + t)dθ on C. By Stokes theorem, F dα 1 = 1. If Ω is any volume form of F which equals to dt dθ in C, then, by de Rham s theorem, Ω dα 1 = dβ, where β is a 1-form which is zero in C. Let α 2 = α 1 + β. Then α 2 satisfies (1) dα 2 is a volume form, and (2) α 2 = (1 + t)dθ in C. The set of 1-forms on F satisfying (1) and (2) above is convex. So we may joint α 2 and h (α 2 ) by a 1-parameter family of 1-forms on F satisfying (1) and (2), and think of this 1-parameter family as a 1-form α on F (h). Since C S 1 is a collar neighborhood of F (h), we may use (t, θ, φ) as the coordinates of this collar neighbor hood of F (h).

17 1.7. EXISTENCE OF CONTACT STRUCTURES IN DIMENSION 3 17 Notice that F (h) is the total space of a F -bundle over S 1. We denote by dφ the 1-form on F (h) which is the pull-back of dφ on S 1. Then dφ dα is a volume form of F (h). Let γ = α + Kdφ. Since γ dγ = K(dφ dα) + α dα, when K is sufficiently large, γ is a contact form on F (h). In the collar neighborhood C S 1 of F (h), γ = (1 + t)dθ + Kdφ. Let θ 1, r, φ 1 be the coordinates of S 1 D 2. We have a contact form β = dθ 1 + r 2 dφ 1 on S 1 D 2. Let f : F (h) S 1 S 1 be given by Then f (β) = (a + b) dθ + (c + d) dφ. ( ) a b f 1 : (θ 1, φ 1 ) (θ, φ), ad bc = ±1. c d Pick a small ɛ > 0 and let t [ ɛ, ɛ], we may now choose a curve (h 1 (t), h 2 (t)) such that (1) (h 1 (t), h 2 (t)) = (1 + t, K) when t 0, and (h 1 ( ɛ), h 2 ( ɛ)) = (a + b, c + d); (2) (h 1 (t), h 2 (t)) and (h 1(t), h 2(t)) are linearly independent for all t. Then the contact form h 1 (t) dθ + h 2 (t) dφ is defined in a thin neighborhood of F (h) in M = F (h) f S 1 D 2 which glues together contact forms α on F (h) and β on S 1 D 2 to give us a contact form on M.

18 18 CHAPTER 1. BASICS

19 Chapter 2 Bennequin 2.1 Introduction Lutz showed in his thesis of 1971 that in every homotopy class of co-oriented tangent hyperplane fields on S 3, there is a contact structure. Lutz s examples are basically these described in Example Since the homotopy classes of co-oriented tangent hyperplane fields on S 3 can be identified with the homotopy group π 3 (S 2 ) = Z, we see that there are infinitely many non-isomorphic contact structures on S 3. When M is of dimension 3, homotopy classes of co-oriented tangent hyperplane fields on M can be identified with homotopy classes of almost contact structures. Indeed, assuming that M is equipped with a Riemannian metric, the complex structure on a cooriented tangent hyperplane field can be given as a count-clockwise rotation of 90 in each hyperplane. And two almost contact structures n M are homotopic iff their underlying cooriented tangent hyperplane fields are homotopic. Thus the isotopic classification problem of contact structures in dimension 3 could be addressed as follows: Is it true that two co-oriented contact structures within the same homotopy classes of co-oriented tangent hyperplane fields necessarily isotopic? For closed 3-dimensional manifolds, due to Gray s theorem (Theorem 1.19), the question above can be reduced to the question of whether two contact structures are homotopic if they are in the same homotopy class of co-oriented tangent hyperplane fields. For open manifolds, due to Gromov s theorem (Theorem 1.21), the question above becomes whether there are homotopic but non-isotopic co-oriented contact structures. These questions were answered by Bennequin in 1982 for S 3 and R 3. Thus, according to Bennequin, there are non-homotopic contact structures on S 3 whose underlying hyperplane fields are homotopic. And there are homotopic but non-isotopic contact structures on R 3. Let C be a contact structure of a 3-dimensional smooth manifold M. Let S be a smooth 2-dimensional surface and f : S M be an immersion. For every point z S, the tangent map T f : T z S T f(z) M is injective. This immersion is in general position with respect to the contact structure C if the points z S where T f(t z S) = C f(z) are all 19

20 20 CHAPTER 2. BENNEQUIN isolated. If the immersion f is in general position, we may get a line field on S away from some isolated points as follows. For a point z where dim(t f(t z S) C f(z) ) = 1, we simply bring back this tangent line to M at f(z) to a tangent line to S at z via T f. We may think of this line field as a singular line field on S. Integral curves of this singular line field on S gives rise to a singular foliation on S, where the leave at an isolated singular point is degenerated to a point. Theorem 2.1. (Bennequin) Let C 0 be the standard contact structure on R 3. For every imbedded disk V in R 3 in general position with respect to C 0, the induced singular foliation on D has no closed leaves except for its singular points. Example 2.2. Let us now consider the contact form α = r sin r dφ + cos r dz on R 3 with cylindrical coordinates (z, r, φ). On the paraboloid z = r 2, the normal vector of the paraboloid is N = z 2r r. So we have α(n) = 2r cos r. Thus, in a small neighbor hood of the the curve r = π on the paraboloid z = r 2, α(n) 0. Therefore, the curve r = π, z = r 2 is a closed leave of the singular foliation on a disk imbedded in R 3. By Theorem 2.1, the contact structure on R 3 determined by this contact form α is not isomorphic to the standard contact structure on R 3. Theorem 2.1 is closely related with the subject of knot theory. If V is a disk imbedded in R 3, the boundary of V, V, is an imbedding of S 1 in R 3 called an unknotted curve, or simply an unknot. The reason is that there are many other imbeddings of S 1 in R 3, called knotted curves, or simply knots, which can not be realized as the boundary of imbedded disks. The curve γ given by r = π, z = r 2 in Example 2.2 is unknotted as well as Legendrian in the contact structure of Example 2.2. Let R α be the Reeb vector field of the contact form α in Example 2.2, let γ = γ + ɛr α be a curve obtained from γ by a small (ɛ) shift in the direction of R α. Give the curve γ an orientation. The γ inherits an orientation from γ. It is easy to check that the Gauss linking number of γ and γ, lk(γ, γ ), is zero. The following theorem says that such a situation never happens in R 3 with the standard contact structure. Theorem 2.3. (Bennequin) Let γ be a unknotted Legendrian curve in R 3 with respect to the standard contact structure C 0. Let γ be obtained from γ by a small shift in the direction of the Reeb vector of C 0. Then lk(γ, γ ) < 0. We can also use Theorem 2.3 to conclude that the contact structure in Example 2.2 is not isomorphic to the standard contact structure. In fact, Theorem 2.1 is a consequence of Theorem 2.3. This claim is not hard to verify.

21 2.2. THE HOPF CONTACT STRUCTURE The Hopf contact structure We shall use the cylindrical coordinates (r, φ, z) in R 3 with the transformation from and to the Cartesian coordinates (x, y, z) given by φ = arctan y/x and r = x 2 + y 2. The standard contact structure C 0 on R 3, under the Cartesian coordinates (x, y, z), is determined by the equation dz y dx = 0. Lemma 2.4. Under the cylindrical coordinates (r, φ, z), the standard contact structure C 0 on R 3 is isomorphic to the contact structure Ĉ0 determined by the contact form dz +r 2 dφ. Proof. We have d(z + xy) (2y) dx = dz + x dy y dx = dz + r 2 dφ. Thus (x, y, z) (x, 2y, z + xy) is the contactomorphism between C 0 and Ĉ0. On each cylinder r = constant, we may solve the equation dz + r 2 dφ = 0 to get its integral curves z = r 2 φ + constant. Exercise 2.5. Show that the integral curve of the Reeb vector field of Ĉ0 passing through (a, b, c) is given by x = ae t, y = be t, z = t + c. The standard contact structure C 0 or Ĉ0 has a more intrinsic definition coming from complex geometry. Let C 2 be equipped with the Hermitian inner product <(z 1, z 2 ), (w 1, w 2 )> = z 1 w 1 + z 2 w 2. Then S 3 is given as the subset of C 2 specified by the equation (z 1, z 2 ) = 1. For each point (z 1, z 2 ) S 3, the complex line L (z1,z 2 ) = {(w 1, w 2 ) ; z 1 w 1 + z 2 w 2 = 0} perpendicular to (z 1, z 2 ) can be thought of as a tangent hyperplane of S 3. Thus we get a tangent hyperplane field on S 3. If we use the polar coordinates, z 1 = r 1 e iφ 1 and z 2 = r 2 e iφ 2, S 3 is given by the equation r r 2 2 = 1 and the tangent hyperplane L (z1,z 2 ) is given by the equation 0 = z 1 d z 1 + z 2 d z 2 = 1 2 d(r2 1 + r 2 2) i(r 2 1 dφ 1 + r 2 2 dφ 2 ) = i(r 2 1 dφ 1 + r 2 2 dφ 2 ). Therefore the tangent hyperplane field L (z1,z 2 ) is a contact structure on S 3 with the contact form r 2 1 dφ 1 + r 2 2 dφ 2. We will call this contact structure on S 3 the Hopf contact structure.

### 8.1 Examples, definitions, and basic properties

8 De Rham cohomology Last updated: May 21, 211. 8.1 Examples, definitions, and basic properties A k-form ω Ω k (M) is closed if dω =. It is exact if there is a (k 1)-form σ Ω k 1 (M) such that dσ = ω.

### Surface bundles over S 1, the Thurston norm, and the Whitehead link

Surface bundles over S 1, the Thurston norm, and the Whitehead link Michael Landry August 16, 2014 The Thurston norm is a powerful tool for studying the ways a 3-manifold can fiber over the circle. In

### Oberwolfach Preprints

Oberwolfach Preprints OWP 2010-01 There is a Unique Real Tight Contact 3-Ball Mathematisches Forschungsinstitut Oberwolfach ggmbh Oberwolfach Preprints (OWP) ISSN 1864-7596 Oberwolfach Preprints (OWP)

### TOPOLOGY OF SINGULAR FIBERS OF GENERIC MAPS

TOPOLOGY OF SINGULAR FIBERS OF GENERIC MAPS OSAMU SAEKI Dedicated to Professor Yukio Matsumoto on the occasion of his 60th birthday Abstract. We classify singular fibers of C stable maps of orientable

### FIXED POINT SETS OF FIBER-PRESERVING MAPS

FIXED POINT SETS OF FIBER-PRESERVING MAPS Robert F. Brown Department of Mathematics University of California Los Angeles, CA 90095 e-mail: rfb@math.ucla.edu Christina L. Soderlund Department of Mathematics

### Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

### Fiber sums of genus 2 Lefschetz fibrations

Proceedings of 9 th Gökova Geometry-Topology Conference, pp, 1 10 Fiber sums of genus 2 Lefschetz fibrations Denis Auroux Abstract. Using the recent results of Siebert and Tian about the holomorphicity

### Mathematical Physics, Lecture 9

Mathematical Physics, Lecture 9 Hoshang Heydari Fysikum April 25, 2012 Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, 2012 1 / 42 Table of contents 1 Differentiable manifolds 2 Differential

### 14.11. Geodesic Lines, Local Gauss-Bonnet Theorem

14.11. Geodesic Lines, Local Gauss-Bonnet Theorem Geodesics play a very important role in surface theory and in dynamics. One of the main reasons why geodesics are so important is that they generalize

### Table of Contents. Introduction... 1 Chapter 1. Vector Bundles... 4. Chapter 2. K Theory... 38. Chapter 3. Characteristic Classes...

Table of Contents Introduction.............................. 1 Chapter 1. Vector Bundles.................... 4 1.1. Basic Definitions and Constructions............ 6 Sections 7. Direct Sums 9. Inner Products

### Singular fibers of stable maps and signatures of 4 manifolds

359 399 359 arxiv version: fonts, pagination and layout may vary from GT published version Singular fibers of stable maps and signatures of 4 manifolds OSAMU SAEKI TAKAHIRO YAMAMOTO We show that for a

### Cabling and transverse simplicity

Annals of Mathematics, 162 (2005), 1305 1333 Cabling and transverse simplicity By John B. Etnyre and Ko Honda Abstract We study Legendrian knots in a cabled knot type. Specifically, given a topological

### The cover SU(2) SO(3) and related topics

The cover SU(2) SO(3) and related topics Iordan Ganev December 2011 Abstract The subgroup U of unit quaternions is isomorphic to SU(2) and is a double cover of SO(3). This allows a simple computation of

### Section 1.1. Introduction to R n

The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to

### tr g φ hdvol M. 2 The Euler-Lagrange equation for the energy functional is called the harmonic map equation:

Notes prepared by Andy Huang (Rice University) In this note, we will discuss some motivating examples to guide us to seek holomorphic objects when dealing with harmonic maps. This will lead us to a brief

### Introduction to Topology

Introduction to Topology Tomoo Matsumura November 30, 2010 Contents 1 Topological spaces 3 1.1 Basis of a Topology......................................... 3 1.2 Comparing Topologies.......................................

### Math 225A, Differential Topology: Homework 3

Math 225A, Differential Topology: Homework 3 Ian Coley October 17, 2013 Problem 1.4.7. Suppose that y is a regular value of f : X Y, where X is compact and dim X = dim Y. Show that f 1 (y) is a finite

### Parametric Curves. (Com S 477/577 Notes) Yan-Bin Jia. Oct 8, 2015

Parametric Curves (Com S 477/577 Notes) Yan-Bin Jia Oct 8, 2015 1 Introduction A curve in R 2 (or R 3 ) is a differentiable function α : [a,b] R 2 (or R 3 ). The initial point is α[a] and the final point

### The Mathematics of Origami

The Mathematics of Origami Sheri Yin June 3, 2009 1 Contents 1 Introduction 3 2 Some Basics in Abstract Algebra 4 2.1 Groups................................. 4 2.2 Ring..................................

### How to minimize without knowing how to differentiate (in Riemannian geometry)

How to minimize without knowing how to differentiate (in Riemannian geometry) Spiro Karigiannis karigiannis@maths.ox.ac.uk Mathematical Institute, University of Oxford Calibrated Geometries and Special

### A QUICK GUIDE TO THE FORMULAS OF MULTIVARIABLE CALCULUS

A QUIK GUIDE TO THE FOMULAS OF MULTIVAIABLE ALULUS ontents 1. Analytic Geometry 2 1.1. Definition of a Vector 2 1.2. Scalar Product 2 1.3. Properties of the Scalar Product 2 1.4. Length and Unit Vectors

### Sets of Fibre Homotopy Classes and Twisted Order Parameter Spaces

Communications in Mathematical Physics Manuscript-Nr. (will be inserted by hand later) Sets of Fibre Homotopy Classes and Twisted Order Parameter Spaces Stefan Bechtluft-Sachs, Marco Hien Naturwissenschaftliche

### The Dirichlet Unit Theorem

Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if

### Two fixed points can force positive entropy of a homeomorphism on a hyperbolic surface (Not for publication)

Two fixed points can force positive entropy of a homeomorphism on a hyperbolic surface (Not for publication) Pablo Lessa May 29, 2012 1 Introduction A common theme in low-dimensional dynamical systems

### Chapter 2. Parameterized Curves in R 3

Chapter 2. Parameterized Curves in R 3 Def. A smooth curve in R 3 is a smooth map σ : (a, b) R 3. For each t (a, b), σ(t) R 3. As t increases from a to b, σ(t) traces out a curve in R 3. In terms of components,

### LECTURE III. Bi-Hamiltonian chains and it projections. Maciej B laszak. Poznań University, Poland

LECTURE III Bi-Hamiltonian chains and it projections Maciej B laszak Poznań University, Poland Maciej B laszak (Poznań University, Poland) LECTURE III 1 / 18 Bi-Hamiltonian chains Let (M, Π) be a Poisson

### Metrics on SO(3) and Inverse Kinematics

Mathematical Foundations of Computer Graphics and Vision Metrics on SO(3) and Inverse Kinematics Luca Ballan Institute of Visual Computing Optimization on Manifolds Descent approach d is a ascent direction

### Solutions for Review Problems

olutions for Review Problems 1. Let be the triangle with vertices A (,, ), B (4,, 1) and C (,, 1). (a) Find the cosine of the angle BAC at vertex A. (b) Find the area of the triangle ABC. (c) Find a vector

### Symplectic structures on fiber bundles

Symplectic structures on fiber bundles François Lalonde Université du Québec à Montréal (flalonde@math.uqam.ca) Dusa McDuff State University of New York at Stony Brook (dusa@math.sunysb.edu) Aug 27, 2000,

### 4. Expanding dynamical systems

4.1. Metric definition. 4. Expanding dynamical systems Definition 4.1. Let X be a compact metric space. A map f : X X is said to be expanding if there exist ɛ > 0 and L > 1 such that d(f(x), f(y)) Ld(x,

### Homotopy groups of spheres and low-dimensional topology

Homotopy groups of spheres and low-dimensional topology Andrew Putman Abstract We give a modern account of Pontryagin s approach to calculating π n+1 (S n ) and π n+2 (S n ) using techniques from low-dimensional

### THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS

THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear

### Math 241, Exam 1 Information.

Math 241, Exam 1 Information. 9/24/12, LC 310, 11:15-12:05. Exam 1 will be based on: Sections 12.1-12.5, 14.1-14.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)

### The Tangent Bundle. Jimmie Lawson Department of Mathematics Louisiana State University. Spring, 2006

The Tangent Bundle Jimmie Lawson Department of Mathematics Louisiana State University Spring, 2006 1 The Tangent Bundle on R n The tangent bundle gives a manifold structure to the set of tangent vectors

### Math 497C Sep 17, Curves and Surfaces Fall 2004, PSU

Math 497C Sep 17, 2004 1 Curves and Surfaces Fall 2004, PSU Lecture Notes 4 1.9 Curves of Constant Curvature Here we show that the only curves in the plane with constant curvature are lines and circles.

### Rotation Rate of a Trajectory of an Algebraic Vector Field Around an Algebraic Curve

QUALITATIVE THEORY OF DYAMICAL SYSTEMS 2, 61 66 (2001) ARTICLE O. 11 Rotation Rate of a Trajectory of an Algebraic Vector Field Around an Algebraic Curve Alexei Grigoriev Department of Mathematics, The

### 1 Local Brouwer degree

1 Local Brouwer degree Let D R n be an open set and f : S R n be continuous, D S and c R n. Suppose that the set f 1 (c) D is compact. (1) Then the local Brouwer degree of f at c in the set D is defined.

### arxiv: v1 [math.mg] 6 May 2014

DEFINING RELATIONS FOR REFLECTIONS. I OLEG VIRO arxiv:1405.1460v1 [math.mg] 6 May 2014 Stony Brook University, NY, USA; PDMI, St. Petersburg, Russia Abstract. An idea to present a classical Lie group of

### RIGIDITY OF HOLOMORPHIC MAPS BETWEEN FIBER SPACES

RIGIDITY OF HOLOMORPHIC MAPS BETWEEN FIBER SPACES GAUTAM BHARALI AND INDRANIL BISWAS Abstract. In the study of holomorphic maps, the term rigidity refers to certain types of results that give us very specific

### Course 221: Analysis Academic year , First Semester

Course 221: Analysis Academic year 2007-08, First Semester David R. Wilkins Copyright c David R. Wilkins 1989 2007 Contents 1 Basic Theorems of Real Analysis 1 1.1 The Least Upper Bound Principle................

### Cylinder Maps and the Schwarzian

Cylinder Maps and the Schwarzian John Milnor Stony Brook University (www.math.sunysb.edu) Bremen June 16, 2008 Cylinder Maps 1 work with Araceli Bonifant Let C denote the cylinder (R/Z) I. (.5, 1) fixed

### Finite covers of a hyperbolic 3-manifold and virtual fibers.

Claire Renard Institut de Mathématiques de Toulouse November 2nd 2011 Some conjectures. Let M be a hyperbolic 3-manifold, connected, closed and oriented. Theorem (Kahn, Markovic) The fundamental group

### Course 421: Algebraic Topology Section 1: Topological Spaces

Course 421: Algebraic Topology Section 1: Topological Spaces David R. Wilkins Copyright c David R. Wilkins 1988 2008 Contents 1 Topological Spaces 1 1.1 Continuity and Topological Spaces...............

### FIBER PRODUCTS AND ZARISKI SHEAVES

FIBER PRODUCTS AND ZARISKI SHEAVES BRIAN OSSERMAN 1. Fiber products and Zariski sheaves We recall the definition of a fiber product: Definition 1.1. Let C be a category, and X, Y, Z objects of C. Fix also

### DIFFERENTIAL GEOMETRY HW 2

DIFFERENTIAL GEOMETRY HW 2 CLAY SHONKWILER 2 Prove that the only orientation-reversing isometries of R 2 are glide reflections, that any two glide-reflections through the same distance are conjugate by

### Lecture 1-1,2. Definition. Sets A and B have the same cardinality if there is an invertible function f : A B.

Topology Summary This is a summary of the results discussed in lectures. These notes will be gradually growing as the time passes by. Most of the time proofs will not be given. You are supposed to be able

### The Topology of Fiber Bundles Lecture Notes. Ralph L. Cohen Dept. of Mathematics Stanford University

The Topology of Fiber Bundles Lecture Notes Ralph L. Cohen Dept. of Mathematics Stanford University Contents Introduction v Chapter 1. Locally Trival Fibrations 1 1. Definitions and examples 1 1.1. Vector

### Geometric Modelling Summer 2016

Geometric Modelling Summer 2016 Prof. Dr. Hans Hagen http://gfx.uni-kl.de/~gm Prof. Dr. Hans Hagen Geometric Modelling Summer 2016 1 Ane Spaces, Elliptic Prof. Dr. Hans Hagen Geometric Modelling Summer

### A new viewpoint on geometry of a lightlike hypersurface in a semi-euclidean space

A new viewpoint on geometry of a lightlike hypersurface in a semi-euclidean space Aurel Bejancu, Angel Ferrández Pascual Lucas Saitama Math J 16 (1998), 31 38 (Partially supported by DGICYT grant PB97-0784

### mathematicae ~) Springer-Verlag 1989

Invent. math. 98, 623-637 (1989) ~nve~ltiones mathematicae ~) Springer-Verlag 1989 Classification of overtwisted contact structures on 3-manifolds Y. Eliashberg*'** Department of Mathematics, University

### SHINPEI BABA AND SUBHOJOY GUPTA

HOLONOMY MAP FIBERS OF CP 1 -STRUCTURES IN MODULI SPACE SHINPEI BABA AND SUBHOJOY GUPTA Abstract. Let S be a closed oriented surface of genus g 2. Fix an arbitrary non-elementary representation ρ: π 1

### 4. Complex integration: Cauchy integral theorem and Cauchy integral formulas. Definite integral of a complex-valued function of a real variable

4. Complex integration: Cauchy integral theorem and Cauchy integral formulas Definite integral of a complex-valued function of a real variable Consider a complex valued function f(t) of a real variable

### Chapter 1. Isometries of the Plane

Chapter 1 Isometries of the Plane For geometry, you know, is the gate of science, and the gate is so low and small that one can only enter it as a little child. (W. K. Clifford) The focus of this first

### Surface Normals and Tangent Planes

Surface Normals and Tangent Planes Normal and Tangent Planes to Level Surfaces Because the equation of a plane requires a point and a normal vector to the plane, nding the equation of a tangent plane to

### Remarks on Lagrangian singularities, caustics, minimum distance lines

Remarks on Lagrangian singularities, caustics, minimum distance lines Department of Mathematics and Statistics Queen s University CRM, Barcelona, Spain June 2014 CRM CRM, Barcelona, SpainJune 2014 CRM

### 11.1. Objectives. Component Form of a Vector. Component Form of a Vector. Component Form of a Vector. Vectors and the Geometry of Space

11 Vectors and the Geometry of Space 11.1 Vectors in the Plane Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. 2 Objectives! Write the component form of

### Proof. The map. G n i. where d is the degree of D.

7. Divisors Definition 7.1. We say that a scheme X is regular in codimension one if every local ring of dimension one is regular, that is, the quotient m/m 2 is one dimensional, where m is the unique maximal

### Solutions to Homework 10

Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x

### Ideal Class Group and Units

Chapter 4 Ideal Class Group and Units We are now interested in understanding two aspects of ring of integers of number fields: how principal they are (that is, what is the proportion of principal ideals

### LINEAR ALGEBRA W W L CHEN

LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,

### ISOMETRIES OF R n KEITH CONRAD

ISOMETRIES OF R n KEITH CONRAD 1. Introduction An isometry of R n is a function h: R n R n that preserves the distance between vectors: h(v) h(w) = v w for all v and w in R n, where (x 1,..., x n ) = x

### Mathematical Methods of Engineering Analysis

Mathematical Methods of Engineering Analysis Erhan Çinlar Robert J. Vanderbei February 2, 2000 Contents Sets and Functions 1 1 Sets................................... 1 Subsets.............................

### Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011

Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely

### State of Stress at Point

State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention,

### Practice Final Math 122 Spring 12 Instructor: Jeff Lang

Practice Final Math Spring Instructor: Jeff Lang. Find the limit of the sequence a n = ln (n 5) ln (3n + 8). A) ln ( ) 3 B) ln C) ln ( ) 3 D) does not exist. Find the limit of the sequence a n = (ln n)6

### INVARIANT METRICS WITH NONNEGATIVE CURVATURE ON COMPACT LIE GROUPS

INVARIANT METRICS WITH NONNEGATIVE CURVATURE ON COMPACT LIE GROUPS NATHAN BROWN, RACHEL FINCK, MATTHEW SPENCER, KRISTOPHER TAPP, AND ZHONGTAO WU Abstract. We classify the left-invariant metrics with nonnegative

### Classification of Bundles

CHAPTER 2 Classification of Bundles In this chapter we prove Steenrod s classification theorem of principal G - bundles, and the corresponding classification theorem of vector bundles. This theorem states

### A CONSTRUCTION OF THE UNIVERSAL COVER AS A FIBER BUNDLE

A CONSTRUCTION OF THE UNIVERSAL COVER AS A FIBER BUNDLE DANIEL A. RAMRAS In these notes we present a construction of the universal cover of a path connected, locally path connected, and semi-locally simply

### The Math Circle, Spring 2004

The Math Circle, Spring 2004 (Talks by Gordon Ritter) What is Non-Euclidean Geometry? Most geometries on the plane R 2 are non-euclidean. Let s denote arc length. Then Euclidean geometry arises from the

### Metric Spaces. Chapter 7. 7.1. Metrics

Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some

### d(f(g(p )), f(g(q))) = d(g(p ), g(q)) = d(p, Q).

10.2. (continued) As we did in Example 5, we may compose any two isometries f and g to obtain new mappings f g: E E, (f g)(p ) = f(g(p )), g f: E E, (g f)(p ) = g(f(p )). These are both isometries, the

### Six questions, a proposition and two pictures on Hofer distance for hamiltonian diffeomorphisms on surfaces

1 Six questions, a proposition and two pictures on Hofer distance for hamiltonian diffeomorphisms on surfaces Frédéric Le Roux Laboratoire de mathématiques CNRS UMR 8628 Université Paris-Sud, Bat. 425

### Lectures notes on orthogonal matrices (with exercises) 92.222 - Linear Algebra II - Spring 2004 by D. Klain

Lectures notes on orthogonal matrices (with exercises) 92.222 - Linear Algebra II - Spring 2004 by D. Klain 1. Orthogonal matrices and orthonormal sets An n n real-valued matrix A is said to be an orthogonal

### Dimension Theory for Ordinary Differential Equations

Vladimir A. Boichenko, Gennadij A. Leonov, Volker Reitmann Dimension Theory for Ordinary Differential Equations Teubner Contents Singular values, exterior calculus and Lozinskii-norms 15 1 Singular values

### Hamiltonian manifolds and moment map.

Hamiltonian manifolds and moment map. Nicole BERLINE and Michèle VERGNE July 4, 2011 Contents 1 Setup of Hamiltonian manifolds 5 1.1 Tangent and normal vector bundle............... 5 1.2 Calculus on differential

### Introduction to Characteristic Classes

UNIVERSITY OF COPENHAGEN Faculty of Science Department of Mathematical Sciences Mauricio Esteban Gómez López Introduction to Characteristic Classes Supervisors: Jesper Michael Møller, Ryszard Nest 1 Abstract

### THE MINIMAL GENUS PROBLEM

THE MINIMAL GENUS PROBLEM TERRY LAWSON Abstract. This paper gives a survey of recent work on the problem of finding the minimal genus of an embedded surface which represents a two-dimensional homology

### A numerable cover of a topological space X is one which possesses a partition of unity.

Chapter 1 I. Fibre Bundles 1.1 Definitions Definition 1.1.1 Let X be a topological space and let {U j } j J be an open cover of X. A partition of unity relative to the cover {U j } j J consists of a set

### Singular fibers of stable maps and signatures of 4 manifolds

Geometry & Topology 10 (2006) 359 399 359 Singular fibers of stable maps and signatures of 4 manifolds OSAMU SAEKI TAKAHIRO YAMAMOTO We show that for a C 1 stable map of an oriented 4 manifold into a 3

### We call this set an n-dimensional parallelogram (with one vertex 0). We also refer to the vectors x 1,..., x n as the edges of P.

Volumes of parallelograms 1 Chapter 8 Volumes of parallelograms In the present short chapter we are going to discuss the elementary geometrical objects which we call parallelograms. These are going to

### CARTAN S GENERALIZATION OF LIE S THIRD THEOREM

CARTAN S GENERALIZATION OF LIE S THIRD THEOREM ROBERT L. BRYANT MATHEMATICAL SCIENCES RESEARCH INSTITUTE JUNE 13, 2011 CRM, MONTREAL In many ways, this talk (and much of the work it reports on) owes its

### 2 Complex Functions and the Cauchy-Riemann Equations

2 Complex Functions and the Cauchy-Riemann Equations 2.1 Complex functions In one-variable calculus, we study functions f(x) of a real variable x. Likewise, in complex analysis, we study functions f(z)

### Recall that the gradient of a differentiable scalar field ϕ on an open set D in R n is given by the formula:

Chapter 7 Div, grad, and curl 7.1 The operator and the gradient: Recall that the gradient of a differentiable scalar field ϕ on an open set D in R n is given by the formula: ( ϕ ϕ =, ϕ,..., ϕ. (7.1 x 1

### 1 if 1 x 0 1 if 0 x 1

Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or

### Invariant Metrics with Nonnegative Curvature on Compact Lie Groups

Canad. Math. Bull. Vol. 50 (1), 2007 pp. 24 34 Invariant Metrics with Nonnegative Curvature on Compact Lie Groups Nathan Brown, Rachel Finck, Matthew Spencer, Kristopher Tapp and Zhongtao Wu Abstract.

### SOME PROPERTIES OF FIBER PRODUCT PRESERVING BUNDLE FUNCTORS

SOME PROPERTIES OF FIBER PRODUCT PRESERVING BUNDLE FUNCTORS Ivan Kolář Abstract. Let F be a fiber product preserving bundle functor on the category FM m of the proper base order r. We deduce that the r-th

### Nonzero degree tangential maps between dual symmetric spaces

ISSN 1472-2739 (on-line) 1472-2747 (printed) 709 Algebraic & Geometric Topology Volume 1 (2001) 709 718 Published: 30 November 2001 ATG Nonzero degree tangential maps between dual symmetric spaces Boris

### Effective homotopy of the fiber of a Kan fibration

Effective homotopy of the fiber of a Kan fibration Ana Romero and Francis Sergeraert 1 Introduction Inspired by the fundamental ideas of the effective homology method (see [5] and [4]), which makes it

### BANACH AND HILBERT SPACE REVIEW

BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but

### 6. Define log(z) so that π < I log(z) π. Discuss the identities e log(z) = z and log(e w ) = w.

hapter omplex integration. omplex number quiz. Simplify 3+4i. 2. Simplify 3+4i. 3. Find the cube roots of. 4. Here are some identities for complex conjugate. Which ones need correction? z + w = z + w,

### The Three Reflections Theorem

The Three Reflections Theorem Christopher Tuffley Institute of Fundamental Sciences Massey University, Palmerston North 24th June 2009 Christopher Tuffley (Massey University) The Three Reflections Theorem

### Figure 1: Volume between z = f(x, y) and the region R.

3. Double Integrals 3.. Volume of an enclosed region Consider the diagram in Figure. It shows a curve in two variables z f(x, y) that lies above some region on the xy-plane. How can we calculate the volume

### MICROLOCAL ANALYSIS OF THE BOCHNER-MARTINELLI INTEGRAL

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 0002-9939(XX)0000-0 MICROLOCAL ANALYSIS OF THE BOCHNER-MARTINELLI INTEGRAL NIKOLAI TARKHANOV AND NIKOLAI VASILEVSKI

### The determinant of a skew-symmetric matrix is a square. This can be seen in small cases by direct calculation: 0 a. 12 a. a 13 a 24 a 14 a 23 a 14

4 Symplectic groups In this and the next two sections, we begin the study of the groups preserving reflexive sesquilinear forms or quadratic forms. We begin with the symplectic groups, associated with

### This makes sense. t 2 1 + 1/t 2 dt = 1. t t 2 + 1dt = 2 du = 1 3 u3/2 u=5

1. (Line integrals Using parametrization. Two types and the flux integral) Formulas: ds = x (t) dt, d x = x (t)dt and d x = T ds since T = x (t)/ x (t). Another one is Nds = T ds ẑ = (dx, dy) ẑ = (dy,

### 7. The Gauss-Bonnet theorem

7. The Gauss-Bonnet theorem 7. Hyperbolic polygons In Euclidean geometry, an n-sided polygon is a subset of the Euclidean plane bounded by n straight lines. Thus the edges of a Euclidean polygon are formed

### ON CERTAIN DOUBLY INFINITE SYSTEMS OF CURVES ON A SURFACE

i93 c J SYSTEMS OF CURVES 695 ON CERTAIN DOUBLY INFINITE SYSTEMS OF CURVES ON A SURFACE BY C H. ROWE. Introduction. A system of co 2 curves having been given on a surface, let us consider a variable curvilinear

### Samuel Omoloye Ajala

49 Kragujevac J. Math. 26 (2004) 49 60. SMOOTH STRUCTURES ON π MANIFOLDS Samuel Omoloye Ajala Department of Mathematics, University of Lagos, Akoka - Yaba, Lagos NIGERIA (Received January 19, 2004) Abstract.