Manifolds obtained by revolving rigid submanifolds


I'd like to know how to find all boundaryless
n-dimensional manifolds (up to topological
equivalence) such that:

1. They contain an boundaryless submanifold
of codimension 1 (i.e. dimension n-1)
whose curvature is everywhere minimal.

2. The submanifold can sweep out the manifold,
without deformation of either, crossing
every point of the manifold exactly once
(i.e. no fixed points or even multiple
crossing points).

Together these seem quite stringent conditions,
which I'm hoping narrow down the possibilities
to one or two topologically distinct manifolds
per dimension.

For example, a circular torus has an infinite
number of closed geodesic curves obtained by
an integer number of twists round the torus.
Furthermore the whole torus can be swept out by
displacing any such twist by a suitable amount,
without crossing any former position through
which it passed during this displacement, but
returning to the starting twist at the end of
the displacement.

I think a Mobius bottle allows a similar
construction. But rotating a circle about
a diameter to create a spherical surface is
an example that does not satisfy the above
conditions, because it has two fixed points.

(The ultimate aim is to find constant curvature
manifolds which allow everywhere locally parallel
vector fields, and such that the manifold can be
traversed, with each point covered for exactly
one position, by a rigid submanifold always
everywhere parallel to the vector field in
the "parent" manifold.)

Would this problem have something to do with
Lie theory by any chance? Any references or
keywords for me to search on the web will be
much appreciated.


Cheers

John R Ramsden (jhnrmsdn@xxxxxxxxxxxx)
remove M from here to reply ^

NOTE: The From email address that shows up
in Google groups is defunct!

.

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