Given three unity vectors from the origin to the surface of a sphere centred on the origin we can get a quantity $(A\times B).C$
If we rotate the sphere by rigid rotations this quantity remains the same.
Are there any groups of diffeomorphisms which keep this quantitiy the same?
It's not so obvious that there aren't but seems likeley that there aren't.
Let $V(u,v,w)=|(u\times v)\cdot w|$. We show that if $f$ is any self map on the sphere satisfying $$V(f(u),f(v),f(w))=V(u,v,w),$$ then $f$ is an isometry of the sphere. Let $i,j,k$ be the standard basis vectors of $\mathbb{R}^3$. For unit vectors $u,v,w$, the quantity is maximum possible (value=1) if and only if the three vectors are mutually perpendicular. Without loss of generality, we can assume $f(k)=k$ (otherwise, compose with an isometry). It follows that $i,j$ are mapped to two perpendicular vectors on the $xy$-plane. By composing with a rotation around the $z$-axis followed by a reflection if necessary, we can assume $f(i)=i$ and $f(j)=j$. Now let $u=(a,b,c)$ be an arbitrary vector and let $f(u)=(a',b',c')$. We have $$(u \times i)\cdot j=(f(u) \times i) \cdot j,$$ which implies that $c=c'$. Similarly one can show $a=a'$ and $b=b'$. This completes the proof.