| 13:30-14:30 |
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Lecture 1. A geometric motivation: Parallel second fundamental form. |
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Abstract:
Any curve with constant nonzero curvature in euclidean plane
is a circle. The subject of our talks is a generalization
to arbitrary dimension and codimension. Curves are replaced
by submanifolds, curvature by the second fundamental form.
Submanifolds with constant (= parallel) second fundamental form
turn out to be objects of fundamental importance in basic mathematics;
they include the matrix groups O(n), U(n), Sp(n) and the Grassmannians
(the set of all vector subspaces of a given dimension). From the
geometric view point they still resemble the planar circle,
being invariant under reflection along each of their normal
spaces ("extrinsic symmetric"). The aim of the first talk is
to indicate a common construction for all these objects: they form
certain orbits of the rotation (isotropy) groups of other symmetric
spaces (= Riemannian spaces with an isometric point reflection
at every point). |
| 14:50-15:50 |
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Lecture 2. Intrinsic and extrinsic geometry. |
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Abstract:
Extrinsic symmetric spaces form a subfamily of symmetric spaces.
What are their special features? Either they carry a parallel
complex structure (Hermitian symmetric spaces) or they are real
forms of such spaces. They are closely related to the so called center
of other symmetric spaces. Their maximal tori are also extrinsic
symmetric and therefore an orthogonal product of planar circles.
All their isometries extend to isometries of the ambient space.
Extrinsic symmetric spaces are embedded not only in euclidean space,
but also in in other symmetric spaces where they are again extrinsic
symmetric. In fact these are the only full extrinsic symmetric
subspaces of symmetric spaces. |
| 16:10-17:10 |
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Lecture 3. The noncompact transformation group. |
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Abstract:
Extrinsic symmetric spaces are compact, but they allow a noncompact
group of transformations extending the isometry group. E.g. for the
round sphere this is the group of conformal transformation, for
projective space it is the projective linear group. This noncompact
group is characteristic for all extrinsic symmetric spaces. It is
generated by the gradient flow of the height functions on the ambient
space. Every compact symmetric space has a noncompact dual symmetric
space; for extrinsic symmetric spaces this is equivariantly embedded as
an open subset of the compact space where the isometry group of the dual
becomes part of the noncompact group on the extrinsic symmetric space. |
