Embedding and Knotting of Positive Curvature Surfaces in 3-space
Herman Gluck
Abstract: In 3-space, compact orientable surfaces with
nonempty boundary and positive curvature play the role of Seifert
surfaces in a curvature- sensitive version of knot theory. The
following result states that the isotopy classes of such surfaces are
in one-one correspondence with the isotopy classes of ordinary
surfaces which have no constraint on their curvature.
THEOREM. (a) In 3-space, any compact orientable surface with
nonempty boundary can be deformed into one with positive curvature.
(b) Any two such surfaces with positive curvature can be
deformed into one another through surfaces of positive curvature if
and only if they can be deformed into one another through ordinary
surfaces, preserving their natural orientations.