Not surprisingly, there are several other approaches to deriving the possible forms of (). Another derivation exploits

Question:

Not surprisingly, there are several other approaches to deriving the possible forms of ∑(χ). Another derivation exploits the symmetries of the Riemann tensor.

(a) The 3-dimensional Riemann curvature tensor of the hypersurface must be homogeneous and isotropic. Explain why it should therefore only involve the metric tensor and not any other tensor, for example, not the Levi-Civita tensor.

(b) Show that the only combination of these quantities that exhibits the full symmetry properties of the Riemann tensor is

where K is a constant.

(c) Write a computer algebra routine to evaluate the Riemann tensor from the metric tensor (28.1) directly. By equating it to Eq. (28.12), show that two differential equations must be satisfied and these are identical to Eqs. (28.5) and (28.6).

(d) Compute the Ricci tensor, and compare it with the metric tensor. Comment.

Equations.