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Math Help - differential geometry help

  1. #1
    Junior Member
    Joined
    Nov 2008
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    54

    differential geometry help

    hi my friends ı have some questions for you;
    1.The standart curvature tensor R,for the unit sphere is parallel.
    2. \kappa_\sigma depends only a \sigma(not on generators)
    3.a)if if n=2,then every metric is an Einstein metric(HİNT: R=K.R_1)
    b)Spaces of constant curvature are all EİNSTEİN spaces(HİNT:same reason)
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  2. #2
    Junior Member
    Joined
    Nov 2008
    Posts
    54

    some new questions ı could not solve

    1-)Define the curvature tensor as a(1,3) tensor and calculate its covariant derivative in the direction of X,where X is a fixed vector field.
    2-)let a subset M of R^4 be given by the equation
    M={ (x_1,x_2,x_3,x_4)\in R^4\mid x_1^2+x_2^2=x_3^2+x_4^2=1} prove that M is a two-dimensional differentiable manifold without displaying an ATLAS.(ı don'T know how to show without ATLAS)
    i just think that M= S^1xS^1 where S^1is 1 manifold hence the cross product is two manifold but how can ı show in the language of mathematic.


    i hope some body help me,please i need some help here.
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