Geodesics on the moduli space of oriented circles in \(\mathbb{S}^{3}\)
The Grassmannian of oriented circles in the conformal \(3\)-sphere carries a natural complex structure. The authors describe this space of circles in terms of a quadric in projective \(4\)-space and equip it with a hermitian metric \(h\) of signature \((2,1)\) that is invariant under the action of the conformal group of \(S^3\). Hence they study the geodesics with respect to the real part \(g=\text{Re}\,h\) of this metric. The two main theorems of the text are Theorem 1.1, which provides a geometric interpretation of the geodesics as, up to trivial cases, families of circles on helicoids in space form subgeometries of conformal geometry; Theorem 1.2, which states that there exist countably many geodesics in the space of circles connecting any two given circles.
scientific article; zbMATH DE number 5911458