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

  1. Home
  2. Homework Library
  3. Mathematics
  4. Topology
  5. 1. a) Suppose that j : M N is an injective immersion between two s...

QuestionQuestion

Transcribed TextTranscribed Text

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 [0] € 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?

Solution PreviewSolution Preview

This material may consist of step-by-step explanations on how to solve a problem or examples of proper writing, including the use of citations, references, bibliographies, and formatting. This material is made available for the sole purpose of studying and learning - misuse is strictly forbidden.

    By purchasing this solution you'll be able to access the following files:
    Solution.zip.

    $100.00
    for this solution

    or FREE if you
    register a new account!

    PayPal, G Pay, ApplePay, Amazon Pay, and all major credit cards accepted.

    Find A Tutor

    View available Topology Tutors

    Get College Homework Help.

    Are you sure you don't want to upload any files?

    Fast tutor response requires as much info as possible.

    Decision:
    Upload a file
    Continue without uploading

    SUBMIT YOUR HOMEWORK
    We couldn't find that subject.
    Please select the best match from the list below.

    We'll send you an email right away. If it's not in your inbox, check your spam folder.

    • 1
    • 2
    • 3
    Live Chats