 # 1. a) Suppose that j : M N is an injective immersion between two s...

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1. a) Suppose that j : M N is an injective immersion between two smooth manifolds. (i) Is it true that j is an embedding? (ii) Assume. in addition, that J is a proper map. Is j is a embedding? b) Is a composition of two smooth embeddings an embedding? 2. Let \T : E M be a smooth rank k: vector bundle. Show that any section f € T(E) is a smooth embedding of M in the total space E. 3. Let M = f-1 (1) where f : R3 R is the function f(x,3,2) - 22 + y - z2 Let S be the sphere of radius 2 centered at the origin of IR³ a) Show that M it S. b) Is M ns a submanifold of R³? 4. Let a = xdy Adz + ydz A da + zdxAdy, B = + zdx&dy and X = your a) Compute the Lie derivative Lxa. b) Compute the Lie derivative LxB. 5. Let 0 = xdy - ydx + dz be a one-form on R³ and D its kernel, i.e., D is the rank two vector bundle over R³ with fiber over p = (x,y,z) given by ker 01p = Dp C T,R3. a) Is 0 ^ d0 a volume form on R³? b) Describe all smooth vector fields X on R³ such that ixA = 0. In other words determine = where a, b and C are smooth functions on R3 such that Xp € Dp for every p E R³. c) Is it true that d0(X,Y) - 0 for every X, Y € T(D)? d) Is D an integrable distribution, i.e., can there be a submanifold S of R3 with TS = D? 6. Let M be a smooth manifold and TM its tangent bundle. a) Show that : R X TM TM defined for t € R and X € TM by is a 1-parameter group of diffeomorphisms of the manifold TM which defines a smooth action of R on TM. b) Is the action proper? c) Find the local expression of the vector field E(X) = d(e'X) € TxTM (§ is a vector field t=0 on TM!) in the standard coordinates (x,y) of TM induced from a coordinate system x on M with y the fiber coordinate. d) Show that § is invariant under the flow Or. 7. Let 22 be a volume form on an oriented manifold M. For a smooth vector field X let div X be the function defined by Lxob = (divX)(2, where Lx is the Lie derivative. Show that if M is compact then SM divXS = famixS. 8. Let = xz³dyAdz. Compute the integral Sss 7,1 where S² is the unit sphere in R3 with equation r²+y+z²=1. = 9. Let  € H2(S"). a) Show that the integral Ssr O A o is a well defined number on i.e., it is constant on the fixed cohomology class. b) Is O A C a volume form of S4?

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