# differential geometry help

• Jan 1st 2009, 12:15 AM
sah_mat
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.$\displaystyle \kappa_\sigma$ depends only a $\displaystyle \sigma$(not on generators)
3.a)if if n=2,then every metric is an Einstein metric(HİNT:$\displaystyle R=K.R_1$)
b)Spaces of constant curvature are all EİNSTEİN spaces(HİNT:same reason)
• Jan 3rd 2009, 04:43 AM
sah_mat
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 $\displaystyle R^4$ be given by the equation
M={$\displaystyle (x_1,x_2,x_3,x_4)\in R^4\mid x_1^2+x_2^2=x_3^2+x_4^2=1$} 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=$\displaystyle S^1xS^1$ where $\displaystyle S^1$is 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.