Math @ Duke

Publications [#243381] of Robert Bryant
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 Bryant, RL, On surfaces with prescribed shape operator,
Results Math. 40 (2001), No. 1 4, 88 121, vol. 40 no. 14
(October, 2001),
pp. 88121, Springer Nature (Dedicated to ShiingShen Chern on his 90th birthday.) [MR2002i:53010], [math.DG/0107083], [doi]
(last updated on 2019/08/18)
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. From classical and more recent work, it is known 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
have been classified and it is known that there are two such families of shape
operators, one depending essentially on three arbitrary functions of one
variable (called Type I in this article) and another depending essentially on
two arbitrary functions of one variable (called Type II in this article).
In this article, these classification results are rederived, with an emphasis
on explicit computability of the space of solutions. It is shown that, for
operators of either type, their realizations by immersions can be computed by
quadrature. Moreover, explicit normal forms for each can be computed by
quadrature together with, in the case of Type I, by solving a single linear
second order ODE in one variable. (Even this last step can be avoided in most
Type I cases.)
The space of realizations is discussed in each case, along with some of their
remarkable geometric properties. Several explicit examples are constructed
(mostly already in the literature) and used to illustrate various features of
the problem.


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