Publications [#10115] of Robert Bryant

Papers Published

1. On surfaces with prescribed shape operator, Results in Mathematics, vol. 40 no. 1--4 (2001), pp. 88--121 (Dedicated to Shiing-Shen Chern on his 90th birthday.) [MR2002i:53010], [math.DG/0107083]
   (last updated on 2010/11/19)

   Author's Comments:
   Please note: The published version is, by mistake, a preliminary version. The correct version is the one posted on the arXiv.

   Abstract:
   The problem of immersing a simply connected surface with a prescribed shape operator is discussed.
   I show that, aside from some special degenerate cases, such as when the shape operator can be realized by a surface with one family of principal curves being geodesic, the space of such realizations is a convex set in an affine space of dimension at most 3. The cases where this maximum dimension of realizability is achieved are analyzed and it is found that there are two such families of shape operators, one depending essentially on three arbitrary functions of one variable and another depending essentially on two arbitrary functions of one variable. The space of realizations is discussed in each case, along with some of their remarkable geometric properties. Several explicit examples are constructed.