2

I've been given this exercise in my self studying differential geometry. Assume $f_1,\dotso,f_4$ be a local basis of vector fields on a manifold $M$, and let $\nu_1,\dotso,\nu_4$ be the dual basis.

Assume that $[f_i,f_j]=p_{ij}\,^kf_k$ and that $$\nabla_{f_i}f_j=c^k\,_{ij}f_k.$$ Then express the $ijkw-$component of the Riemann Tensor in terms of the structure constants and the commutator coefficients, with the convention that $$R^w\,_{ijk}=\langle R(f_i,f_j)f_k,\nu^w\rangle.$$

My work: $$ R(f_i,f_j)f_k=[\nabla_{f_i},\nabla_{f_j}]f_k-\nabla_{[f_i,f_j]}f_k$$ Then $$\nabla_{f_i}\nabla_{f_j}f_k=\nabla_{f_i}(c^q\,_{jk}f_q)=f_i(c^q\,_{jk})f_q+c^q\,_{jk}c^l\,_{iq}f_l. $$ $$-\nabla_{f_j}\nabla_{f_i}f_k=-\nabla_{f_j}(c^q\,_{ik}f_q)=-f_j(c^q\,_{ik})f_q-c^q\,_{ik}c^l\,_{jq}f_l. $$ $$\nabla_{[f_i,f_j]}f_k=p_{ij}\,^qc^l\,_{qk}f_l.$$

Then pairing with $\nu^w$ I got $$R^w\,_{ijk}=f_i(c^w\,_{jk})+c^q\,_{jk}c^w\,_{iq}-f_j(c^w\,_{ik})-c^q\,_{ik}c^w\,_{jq}-p_{ij}\,^qc^w\,_{qk}.$$

Am I right or am I missing something? thanks for your kindness but I'm a newbie and I would like an hand.

P.S. Einstein summation is intended throughout the post

-Guido-

guido giuliani
  • 1,195
  • The problem is that $\nabla_{f_a}f_b$ is a vector as well as $\nabla_{f_c}(\nabla_{f_a}f_b)$, while you write this last terms as $...f_n f_m+...f_r$, which is no vector (it is a 2-vector added to a vector). I will write an answer, as soon as I have time, ok? – Avitus Aug 15 '13 at 15:06
  • I don't see any $f_nf_m$ in his calculation. The term $f_i(c^q_{jk})f_q$ involves differentiation of $c$ in the direction of $f_i$ rather than product. – Mikhail Katz Aug 15 '13 at 15:14
  • aren't the c's constants? – Avitus Aug 15 '13 at 15:19

1 Answers1

2
  • Edit: constant $c$'s. Waiting for the OP to confirm/disprove.

Let

$$\nabla_{f_i}(\nabla_{f_j} f_k)=\nabla_{f_i}(c^k_{jk}f_k)=c^k_{jk}c^q_{is}f_q, $$

$$\nabla_{f_j}(\nabla_{f_i} f_k)=\nabla_{f_j}(c^t_{ik}f_t)=c^t_{ik}c^r_{jt}f_r, $$

and

$$\nabla_{[f_i,f_j]}( f_k)=p_{ij}^d c^n_{dk} f_n.$$

So

$$\langle R(f_i,f_j)f_k ,\mu^\omega\rangle = \langle c^k_{jk}c^q_{is}f_q +c^t_{ik}c^r_{jt}f_r-p_{ij}^d c^n_{dk} f_n ,\mu^\omega\rangle =c^k_{jk}c^\omega_{is}+c^t_{ik}c^\omega_{jt}-p_{ij}^d c^\omega_{dk}, $$

as $\langle f_i,\mu^\omega\rangle=\delta^\omega_{i}$ for all $ i,\omega=1,2,3,4.$

  • $c$'s are not constant

In this case OP's computations are correct.

Avitus
  • 14,018