A local classification of pseudo-parallel submanifolds of space forms with flat normal bundle, Guillermo Lobos (UFSCar)

In 1999, a pseudo-parallel submanifold of a space form was introduced by Asperti-Lobos-Mercuri as an extrinsic analogue of a pseudo-symmetric manifold in the sense R. Deszcz and as a direct generalization of a semi-parallel submanifold. In this talk we give a classification of pseudo-parallel submanifolds with flat normal bundle of space forms, extending the classification by Dillen-Nölker and Lumiste for the semi-parallel case. This is joint work with Ruy Tojeiro (UFSCar, Brazil).

Topology of positively curved manifolds, Wolfgang Ziller (UPenn)

There are few examples of manifolds with positive sectional curvature, in particular there are only two infinite families, one in dimension 7 due to Eschenburg, and one in dimension 13 due to Bazaikin. I will review some known results concerning topological properties of the Eschenburg spaces and discuss some recent joint work with Luis Florit on topological properties of the Bazaikin spaces.

Conformal immersions of warped products, Ruy Tojeiro (UFSCar)

We prove a general decomposition theorem for conformal immersions into Euclidean space of a warped product of dimension n≥3 of Riemannian manifolds. In particular, we determine all conformal representations of Euclidean space of dimension n≥3 as a warped product of Riemannian manifolds. As a consequence, we classify the conformally flat warped products.

Homogeneous G-structures on affine manifolds and affine immersions, Daniel Tausk (USP)

Let (M,∇) be an affine differentiable manifold endowed with a G-structure P. We say that (M,∇,P) is infinitesimally homogeneous if the curvature and torsion of ∇ and the inner torsion of P are all constant on referentials of M belonging to P. Examples of infinitesimally homogeneous affine manifolds with G-structure are Riemannian (or semi-Riemannian) manifolds with constant sectional curvature and Kähler manifolds with constant holomorphic curvature. We present a result giving necessary and sufficient conditions for the existence of affine G-structure preserving immersions with prescribed second fundamental form and normal connection into infinitesimally homogeneous affine manifolds with G-structure.

Aspectos variacionais relacionados com métricas de Einstein invariantes nas variedades-bandeira, Caio Negreiros (Unicamp)

Iremos nesta palestra discutir como o entendimento da geometria Hermitiana invariante das variedades-bandeira está intimamente relacionado com o estudo das aplicações harmônicas com valores nestes espaços. Discutiremos a estabilidade do funcional energia para uma classe bem grande de aplicações com valores em variedades-bandeiras, sendo dado ênfase no caso em que as variedades-bandeira estão munidas de métricas Kähler-Einstein ou simplesmente são Einstein.

Polar actions on compact rank one symmetric spaces are taut, Claudio Gorodski (USP)

The theory of polar actions can be seen as a general theory of canonical forms (Palais-Terng). The concept of taut submanifold is due to Carter and West in the case of Euclidean ambient and to Terng and Thorbergsson in the general case, and can be traced backed to the work of Chern and Lashof on immersions with minimal total absolute curvature and the subsequent reformulation of that work by Kuiper in terms of critical point theory. After recalling the definitions of polar and taut actions, we will explain our result that a polar action of a compact Lie group on a compact rank one symmetric space is taut with respect to Z₂-coefficients. (Joint work with Leonardo Biliotti.)

Hipersuperfícies Lorentzianas relacionadas a curvas planas, Martha Dussan (USP)

Em 1996, U. Hertrich-Jeromin estabeleceu que as hipersuperfícies conformemente planas na esfera Sⁿ estão relacionadas a equações de curvas planas, as quais são conhecidas por constituir um sistema integrável. Este último fato permite que as ditas hipersuperfícies possam ser estudadas no contexto dos sistemas integráveis e que uma nova visão delas possa ser estabelecida. Nesta direção, ele também provou resultados clássicos de Cartan de classificação local para essas hipersuperfícies. Nesta palestra, apresentarei resultados de um trabalho em conjunto com Prof. M. Magid onde provamos que, similarmente como acontece no caso definido positivo, hipersuperfícies conformemente planas Lorentzianas estão também relacionadas a curvas planas.

Lower bounds for the eigenvalues of the Dirac operator and the Laplacian, Marcos Jardim (Unicamp)

Let M be a riemannian manifold, and consider a complex vector bundle E over X, provided with a connection ∇. We will give a few results concerning lower bounds for the smallest eigenvalue of the Dirac operator coupled to ∇, generalizing known results for the case of E being the tangent bundle TM and ∇ being the Levi-Civitta connection. We will also discuss lower bounds for the spectrum of the coupled Laplacian. (Joint work with Rafael Leão.)

Transformation of periodic instantons, Benoit Charbonneau (McGill)

There is, in principle, a correspondence between solutions to the Yang-Mills-anti-self-dual equation on a quotient of euclidian space and solutions to an associated equation on a dual quotient. In this talk, we will study the Nahm Transform heuristic, and see how this transform associates to an instanton on R x T³ a singular monopole on T³, and what is the precise behaviour at the singularities.

O problema de Björling em grupos de Lie tridimensionais, Irene Onnis (Unicamp)

Daremos um teorema de existência e unicidade para o problema de Björling no caso de um grupo de Lie de dimensão três e falaremos de problemas em aberto que estão relacionados com este assunto.

The Cauchy problem for improper affine spheres and the Hessian-1 equation, Rosa Chaves (USP)

We give a conformal representation for improper affine spheres which is used to solve the Cauchy problem for the Hessian-1equation. With this representation, we characterize the geodesics of an improper affine sphere, study its symmetries and classify the helicoidal ones. Finally, we obtain the complete classification of the isolated singularities of the Hessian-1 Monge-Ampère equation.

Volume of flows and index on unit spheres, Fabiano Gustavo Braga Brito (USP)

We establish a geometric-topological inequality relating the volume of a unit vector field (geometry) and its total index (topology).

On the gradient group of an immersion, Luiz San Martin (Unicamp)

The gradient group Grad(M) of a manifold immersed in an Euclidian space is the group (or rather the pseudo-group) generated by the flows of the gradient vector fields of the height functions. In this talk we discuss the problem of classifying the immersions for which Grad(M) is a finite dimensional (Lie) group. We present several examples and preliminary results towards this classification.