Möbius transformations and isoclinal sequences of spheres
The author takes up anew the problem of isoclinal sequences of spheres in inversive n-space. However, instead of considering (n-1)-spheres, n-balls \(B_ k\) whose bordering spheres \(\partial B_ k\) have common inclination are examined. A sequence \(B_ 0,...,B_{n+1}\) of \(n+2\) equally inclined n-balls in inversive n-space can be transformed into a similar sequence such that the new n-balls are congruent n-spherical caps with common angular radius centered at the vertices of an \((n+1)\)-simplex inscribed in the n-sphere \(\Sigma^ n\) (which is a model for inversive n-space). From this representation it easily follows that the common inclination \(\gamma\) of such a sequence of n-balls is \(\leq -1/(n+1)\); in the case of equality here is essentially one such sequence, but if \(\gamma <-1/(n+1)\) there are two. In the last case the author constructs a doubly infinite sequence of n-balls \(B_ k\) with the property that any \(n+2\) consecutive n-balls in the sequence have the same common inclination. Since n-balls are represented by \((n+2)\)-vectors, this gives rise to a Lorentz transformation M of \({\mathbb{R}}^{n+2}\) such that \(B_ k=B_ 0M^ k\), \(k\in {\mathbb{Z}}\). But either M or -M represents a Möbius transformation M' of the inversive n-space (in any case M' maps \(\partial B_ k\) onto \(\partial B_{k+1})\). For the values \(\gamma \leq -1/(n+1)\), \(\gamma\) \(\neq -1/n\), the author determines the conjugacy class of M' by considering the characteristic equation of M, and it is clarified wether \(M'=M\) or \(M'=-M.\) Finally it is shown how the case \(n=1\), which has to be excluded in previous studies on this problem, fits into the general theory.
isoclinal sequences of spheres in inversive n-space
scientific article; zbMATH DE number 3949150