Page 169 - Intro to Tensor Calculus

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I 11. Verify the equation (1.5.39) which shows that the normal curvature directions are orthogonal. i.e.
verify that Gλ 1 λ 2 + F(λ 1 + λ 2 )+ E =0.

δ
I 12. Verify that δ βγ ωα R ωαβγ =4R λνστ .
στ λν
I 13. Find the first fundamental form and unit normal to the surface defined by z = f(x, y).
I 14. Verify
A i,jk − A i,kj = A σ R σ
.ijk
where
∂ σ ∂ σ n σ n σ
σ
R .ijk = − + − .
∂x j ik ∂x k ij ik nj ij nk
which is sometimes written

s s
∂ ∂

R injk = ∂x j ∂x k + nj nk

[nj, k][nk, i] [ij, s] [ik, s]
I 15. For R ijkl = g iσ R σ show
.jkl

∂ ∂ s s
R injk = [nk, i] − [nj, i]+ [ik, s] − [ij, s]
∂x j ∂x k nj nk
which is sometimes written

n n
∂ ∂

∂x j ∂x k ij
ik
σ
R .ijk = +

σ σ σ σ

ij ik nk nj

I 16. Show
2 2 2 2
1 ∂ g il ∂ g jl ∂ g ik ∂ g jk αβ
R ijkl = − − + + g ([jk, β][il, α] − [jl, β][ik, α]) .
j
i
j
i
2 ∂x ∂x k ∂x ∂x k ∂x ∂x l ∂x ∂x l
I 17. Use the results from problem 15 to show
(i) R jikl = −R ijkl , (ii) R ijlk = −R ijkl , (iii) R klij = R ijkl
Hence, the tensor R ijkl is skew-symmetric in the indices i, j and k, l. Also the tensor R ijkl is symmetric with
respect to the (ij)and (kl) pair of indices.
I 18. Verify the following cyclic properties of the Riemann Christoffel symbol:
(i) R nijk + R njki + R nkij = 0 first index fixed

(ii) R injk + R jnki + R knij = 0 second index fixed
(iii) R ijnk + R jkni + R kinj = 0 third index fixed
(iv) R ikjn + R kjin + R jikn = 0 fourth index fixed

I 19. By employing the results from the previous problems, show all components of the form:
R iijk , R injj , R iijj , R iiii , (no summation on i or j) must be zero.