
Isoparametric submanifolds 


An important topic in my current research is, jointly with
Ernst Heintze, a contribution to the conjecture that isoparametric submanifolds of codimension at least two in Hilbert space are essentially principal
orbits of isotropy representations of symmetric spaces of affine KacMoody
type.
A submanifold of an Euclidean space is called isoparametric
if the eigenvalues of a locally defined
parallel normal vector field are constant and its normal bundle is flat.
In codimension one, the second condition is void
and thus isoparametric hypersurfaces are just those having
constant principal curvatures, namely those with the
simplest local invariants (in particular, homogeneous
hypersurfaces are always isoparametric). The subject had its origins
in geometrical optics around the turn of the 20th century,
but the main early contributions were made by the great É. Cartan
during 193040. The subject was then forgotten for about 30 years
when it was revived by the successive works of Nomizu, Terng, Harle,
Carter and West and many others.
Homogeneous examples are supplied by the principal orbits of isotropy
representations of Riemannian symmetric spaces. On the other hand,
a theorem by Thorbergsson asserts that homogeneity is necessary if
the codimension is different from two (inhomogeneous examples in
codimension two have been constructed by Ozeki and Takeuchi,
and Ferus, Karcher and Münzner). This result, combined with a
remark by Palais and Terng and a classification theorem of
Dadok, implies that in codimension different from two
there are no further examples besides those produced by isotropy
representations of symmetric spaces.
Hence we see a close relation between isoparametric submanifolds
and symmetric spaces.
Further, in a similar way as symmetric spaces
are test/model spaces in Riemannian geometry, also
isoparametric submanifolds are test/model spaces in submanifold geometry.
Isoparametric submanifolds can also be considered in Hilbert space.
By assumption, they are proper Fredholm, which essentially means
that they have finite codimension and compact (selfadjoint) shape operators.
Homogeneous examples are now provided by the isotropy
representations of symmetric spaces of affine KacMoody type.
Further, Thorbergsson's result has been generalized by Heintze and Liu to yield
that homogeneity is necessary if the codimension is different from one and,
similarly to the above,
it is conjectured that in codimension different from one all
examples come from symmetric spaces of affine KacMoody type.
However, since there are neither suitable classifications of HilbertLie groups
nor of their representations, one cannot repeat the strategy
that works finite dimensions. Our work in progress relies on the
geometry of isoparametric submanifolds to attain results
about their classification.

Geometry of orbit spaces 


Together with Alexander Lytchak, we have investigated
representations of compact Lie groups that have isometric orbit
spaces and found invariants and a geometrical description of classes
containing representations with very different
algebraic porperties.
For an orthogonal representation ρ:G→O(V)
of a compact (possibly disconnected)
Lie group G, the quotient metric space V/G is the
most important invariant of the action, at least from the metric
point of view. Moreover,
from the geometrictopologic point of view, some properties of the
action can be recovered from the transverse geometry of the
orbital foliation.
We call two such representations
quotientequivalent if they have the same
orbit space; and one of them a reduction of the other
if the former has strictly smaller dimension.
It is an interesting problem to understand
when two actions can have the same orbit space, or, to find elements of
minimal dimension in a given quotientequivalence class
(minimal reductions). On one hand, the existence
of a reduction implies some, probably severe, restrictions
on the original representation. On the other hand, while most
representations do not admit reductions, many geometrically
interesting ones do. For instance, it is the case for many representations
of low cohomogeneity and for the important class of
polar representations. These are precisely the representations
admitting reductions to finite groups, the simplest example being
that of the Weyl group associated to the adjoint representation
of a compact connected Lie group. Another class is built by the representations
with nontrivial principal isotropy groups, and, generalizing
both previous examples, by representations with
nontrivial copolarity.




Prospective (under)graduate students 


Tenho disposição para orientar alunos que se interessem pelas
seguintes áreas de pesquisa:

Geometria Riemanniana: ações isométricas,
grupos de Lie e espaços simétricos,
geometria de subvariedades (iniciação científica, mestrado
e doutorado)

Representações de grupos de Lie (IC e M)

O método do "repère mobile" ou
referencial móvel (IC e M)

Geometria complexa: superfícies de Riemann
e curvas algébricas, geometria de Kähler
(IC e M)

Geometria algébrica complexa (IC e M)

Geometria métrica (IC e M)

Outros tópicos interessantes (IC e M)

Graduate students (there is no category such as "former graduate students") 



Supervised PostDocs 






