Lie applicable surfaces

G-deformations of maps into projective space

“…The following proposition was proved in [15] in the case that (s, t) = (4, 2). Using analogous arguments one can show that it holds in the case that (s, t) = (3, 3) as well.…”

Lie applicable surfaces and curved flats

“…In [8,17] it is shown thatf is also a Lie applicable surface and that f is an m-Darboux transform off . Thus we say that ( f,f ) is an m-Darboux pair.…”

“…Blaschke [3] showed that together these surfaces are the applicable surfaces of Lie sphere geometry. In [12,16] it is shown that these surfaces constitute an integrable system and this is given a gauge theoretic interpretation in [8,17]. This gives rise to a Lie-geometric analogue of the Darboux transformation for these surfaces.…”

Lie applicable surfaces and curved flats

“…We call f an m-Darboux transform of f . In [8,17] it is shown that f is also a Lie applicable surface and that f is an m-Darboux transform of f . Thus we say that (f, f ) is an m-Darboux pair.…”

“…Blaschke [3] showed that together these surfaces are the applicable surfaces of Lie sphere geometry. In [12,16] it is shown that these surfaces constitute an integrable system and this is given a gauge theoretic interpretation in [8,17]. This gives rise to a Lie-geometric analogue of the Darboux transformation for these surfaces.…”