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.


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.