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I am reading the paper "On Local Loops in Affine Manifolds" by Kikkawa (open access), in which the author considers a manifold with affine connection $(M,\nabla)$. He then builds up a structure called local loop and follows to prove that if such loop has some property (*), and the connection is symmetric (torsion-free), then the Riemann curvature tensor $R$ vanishes at each point (Theorem 2). I have a couple of questions:

(1) How does one get to eq. (7), namely $$R_p(X_p,Y_p)Z_p+R_p(Z_p,Y_p)X_p=0$$

(2) How does eqs. (7) and (8) imply $$R_p(X_p,Y_p)Z_p=0,\;\text{for all}\;X_p,Y_p,Z_p\in T_pM.$$

I have tried using some known identities of the curvature tensor, but with no luck. Any help would be appreciated.


(*) I believe the construction of the local loops and the needed property are irrelevant for this question, since apparently, for Theorem 2 at least, they are only used to obtain relations on the geometry, namely eqs. (1-8). I reckon it is unnecessary to list such equations, since they are available in the linked paper.

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Substituting $X$ with $X+Z$ in $$R(X,Y)X = 0 \tag 6$$ yields \begin{align} 0 &= R(X+Z,Y)(X+Z)\\ &= R(X,Y)X+R(Z,Y)X+R(X,Y)Z+R(Z,Y)Z. \end{align}

Now applying $(6)$ again shows the first and last terms vanish, so we are left with the desired $$R(X,Y)Z+R(Z,Y)X = 0. \tag 7$$

(This argument might be familiar from linear algebra if you rewrite it in terms of the bilinear form $B(X,Z) = R(X,Y)Z.$)

If we now rewrite the Bianchi identity $(8)$ using antisymmetry as

$$ 0 = R(X,Y)Z + R(Z,X)Y - R(Z,Y)X $$

and apply $(6)$ to the last two terms, we get

$$0=R(X,Y)Z - R(Y,X)Z +R(X,Y)Z = 3R(X,Y)Z$$ as desired.

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