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
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
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
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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