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Use 3D lattice and vector field(s) to represent curved manifold?

  1. Jan 22, 2010 #1
    I would like to try and map a small piece of a 3 dimensional curved manifold using a flat 3 dimensional space, and a vector field. Will the following work?

    Take a 3 dimensional cube of size a*a*a that lies in a 6 dimensional space, R^6, with coordinates x1,x2,x3,x4,x5,x6. Let this cube be a regular lattice of N*N*N points. Say one corner of the cube was at the origin. Let the cube be rotated so that three edges of the cube lined up with the x1, x2, and x3 axes. Now let the points of the cube move and deform an infinitesimal amount in the x4, x5, and x6 directions. With such freedom can we represent any small piece of a curved 3 dimensional manifold (some restrictions may apply?)?

    If so it seems we could use a three dimensional vector field to map a small piece of some curved 3 dimensional manifold? A three dimensional vector field could represent the change in coordinates x4, x5, and x6 of each point of the cube.

    Or must points be free to move in all 6 dimensions so that we might represent a curved three dimensional manifold?

    If this is true then can a vector field in our flat three dimensional space represent some curved 3 dimensional manifold (or do we need a pair of vector fields?)?

    Thanks for any help!
  2. jcsd
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