# Calculation of |[X,Y]^V|

I want to follow the proof of Theorem 3.1 in "On Eschenburg's Habilitation on Biquotients" - Wolfgang Ziller.

The situation is as follows: $Q$ is a biinvariant metric on $G$. So from the symmetric tensor $P : T_eG \rightarrow T_eG$, we have a left invariant metric $g\doteq Q(\cdot , P\cdot )$.

Consider the bigquotient $(G,g)// U$. Here we want to calculate

$sec(x,y) = sec_G(X,Y) + \frac{3}{4} | [ X,Y]^V|^2$ where $X$ and $Y$ are hozontal lifts of $x$ and $y$.

To calculate $| [ X,Y]^V|^2$, define one-form $\omega (A) \doteq g ( A, X_L\cdot p + p\cdot X_R)$ where $X_L\cdot p + p\cdot X_R$ is a vertical vector at $p\in G$

If we let $X$ and $Y$ to be horizontal vector fields, let $A$ and $B$ are left invariant vector fields such that $A(p) = X(p)$ and $B(p) = Y(p)$

Then we have $A \omega (B) - B \omega (A) - \omega( [ A,B]) = - \omega([X,Y])$

From $Z_p Q(W, Ad_{p^{-1}} V) = Q(W, - ad_u Ad_{p^{-1}} V)$ where $Z_p$ is a left invariant vector field wrt $g$ and $V$, $W\in T_eG$ (See 404 page in "An exotic sphere with nonnegative sectional curvature"- Gromoll and Meyer), we have

$| \omega([X,Y]) | = | g (Ad_{p^{-1}} X_L, L(A,B) ) - g(X_R,[A,B])|$ where $L(A,B) \doteq - (ad_A)^\ast B + (ad_B)^\ast A - [A,B]$

Question : But in the paper, $L(A,B) = (ad_A)^\ast B - (ad_B)^\ast A - [A,B]$. I do not know where there exists a mistake. Please, help me.

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I haven't read either paper your reference in enough detail to comment on your actual question, but I know that when Gromoll and Meyer say that the $0$ curvature points in $\Sigma^7$ are of smaller dimension, they are wrong. See F. Wilhelm: An exotic sphere with positive curvature almost everywhere, J. Geom. Anal. 11 (2001), 519 - 560 for more details (I currently don't have access to the paper). – Jason DeVito Aug 3 '12 at 8:06
On second thought, here's Wilhelm's paper, from his own website mathdept.ucr.edu/pdf/publications/ladder/11-GM-alm.pdf – Jason DeVito Aug 3 '12 at 8:10