Certain conditions for a Riemannian manifold to be isometric with a sphere:Dedicated to Professor Kentaro Yano on his fiftieth birthday

Lie groupoids and lie algebroids in differential geometry

Lie Groups and Lie Algebras

▶ Examines global problems of geometry and analysis ▶ Looks at interactions between differential geometry and global analysis and their application to problems of theoretical physics ▶ Contributes to an enlargement of the international exchange of research results in the field ▶ Comprehensive coverage, from complex manifolds and related results from complex analysis and algebraic geometry to Lie transformation groups and harmonic analysis ▶ 100% of authors who answered a survey reported that they would definitely publish or probably publish in the journal again

Annals of Global Analysis and Geometry

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About Pure and Applied Mathematics

1: Basic results on Lie groups.- 2: Lie groups with bi-invariant metrics.- 3: Proper and isometric acions.- 4: Adjoint and conjugation actions.- 5: Polar foliations.- 6: Low cohomogeneity actions and positive curvature.- Appendix: Rudiments of smooth manifolds.

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Lie Groups and Geometric Aspects of Isometric Actions

We study existence and non-existence of constant scalar curvature metrics conformal and arbitrarily close to homogeneous metrics on spheres, using variational techniques. This describes all critical points of the Hilbert–Einstein functional on such conformal classes, near homogeneous metrics. Both bifurcation and local rigidity type phenomena are obtained for 1-parameter families of U(n + 1), Sp(n + 1) and Spin(9)-homogeneous metrics.

Bifurcation and local rigidity of homogeneous solutions to the Yamabe problem on spheres

Let g_t be a family of constant scalar curvature metrics on the total space of a Riemannian submersion obtained by shrinking the fibers of an original metric g, so that the submersion collapses as t approaches 0 (i.e., the total space converges to the base in the Gromov-Hausdorff sense). We prove that, under certain conditions, there are at least 3 unit volume constant scalar curvature metrics in the conformal class [g_t] for infinitely many t's accumulating at 0. This holds, e.g., for homogeneous metrics g_t obtained via Cheeger deformation of homogeneous fibrations with fibers of positive scalar curvature.

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Multiplicity of solutions to the Yamabe problem on collapsing Riemannian submersions

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We study existence and non-existence of constant scalar curvature metrics conformal and arbitrarily close to homogeneous metrics on spheres, using variational techniques. This describes all critical points of the Hilbert-Einstein functional on such conformal classes, near homogeneous metrics. Both bifurcation and local rigidity type phenomena are obtained for 1-parameter families of U(n+1), Sp(n+1) and Spin(9)-homogeneous metrics.

Renato G. Bettiol

Papers

Lie Groups and Geometric Aspects of Isometric Actions

Bifurcation and local rigidity of homogeneous solutions to the Yamabe problem on spheres

Multiplicity of solutions to the Yamabe problem on collapsing Riemannian submersions

Multiplicity of solutions to the Yamabe problem on collapsing Riemannian submersions

Bifurcation and local rigidity of homogeneous solutions to the Yamabe problem on spheres