- Lower Bound for Contraction Constants of Non-zero Degree Mappings onto Spheres,
with E. Ciriza, Differential Geom. Appl.,
14/2, 209-216, 2001.
This paper studies contraction constants of non-zero degree mappings from
compact spin Riemannian manifolds onto the standard Riemannian sphere. Assuming uniform
lower bound for the scalar curvature, we find a sharp lower bound for the dilation
constants in terms of the dimension of sphere. In the best case, we prove rigidity.
- Sharp Estimates and the Dirac Operator, Math. Ann., 310,
55-71, 1998.
Article
- Scalar Curvature Estimates for (n+4k)-dimensional Spin Manifolds,
Differential Geom. Appl., 6, 321-326, 1996.
Article
Combing vanishing arguments with certain Gromov-Lawson techniques some
sharp estimates on the scalar curvature are found. The notion of Â-degree of a map is
needed to establish the results in dimension n+4k. The main result states that any compact
spin manifold M, of dimension n+4k, which admits a distance decreasing nonzero Â-degree
map f: M ---> Sn onto the n dimensional sphere must have a point x with
scalar curvature k(x) < n(n-1)/(n+4k)(n+4k-1), otherwise, f is an isometric submersion.
The result is extended to the non-compact case provided the map f is constant at infinity.
- Classical Semisimple Connected Lie Groups of Split Rank One, Center for
Research, College of Science and Health, WPC, T.R. 104, 1996.
This paper presents a detailed study of the connected semisimple classical
Lie groups of split rank 1. A Lie group is an abstract group together with a
differentiable structure that makes the function G x G ----> G given by (x,y) |---->
xy-1 differentiable. Every Lie group has an associated Lie algebra that it is
identify with the tangent space at the identity of G and the algebra multiplication is the
usual bracket, [X,Y]. For a fixed X, we have the endomorphism ad X: g ---> g,
defined by ad X(Z) = [X,Z] . This gives rise to a bilinear form B, called the Killing form
of g, and defined by B(X,Y) = tr(adX.adY). The Lie algebra is said to be semisimple
if B is non-singular. The paper is divided in two parts. In the first part; the groups G
[SO(n,1), SU(n,1) and SP(n,1)] are introduced and some of their topological properties are
derived. Then their Lie Algebras g [so(n,1), su(n,1), sp(n,1)] and their
Killing forms are explicitly calculated. For each group G , a maximal compact subgroup K
is found. The Lie algebra k, of K is a Cartan subalgebra of g and determines a
Cartan decomposition of g, which in turn, gives a decomposition of g into
the root subspaces with respect to k.
- Irreducible Decomposition of L2 Functions on Hopf Bundles,
Linear and Multilinear Algebra, Vol 41, 113-135, 1996.
Article
This article gives explicit decompositions of the spaces L2(Sdn-1)
for d = 1,2,4, into irreducible k-modules.
This is done by letting the groups SO(n,1), SU(n,1) and SP(n,1) act on Sdn-1
for d = 1, 2, 4, respectively and finding Cartan subalgebras of the corresponding
Lie algebras associated to the Lie groups, namely, so(n,1), su(n,1), sp(n,1). Then
by considering the Laplacian and Casimir operators in polar coordinates, the decomposition
of L2(Sn) arises as the eingenspaces of the harmonic functions when
restricted to Sn and as the eigenspaces of the Casimir operator.
- The Scalar Curvature and Volume of a Riemannian Manifold, Geometriae
Dedicata, 56: 1-4, 1995.
Article
The link between volume and scalar curvature of a Riemannian manifold does
not have such an easy description. In dimension two, for example, in the case of he
standard sphere, the classical Gauss-Bonnet theorem relates the two concepts, i.e. the
total curvature of the sphere is 4 times the volume of the sphere. However, in higher
dimension any relation that the volume and scalar curvature may have, requires a finer
analysis. This paper explores the relation between the two concepts from the viewpoint of
contracting maps while provides counterexamples to the use of weaker hypotheses in the
main theorem proved in that paper Sharp Estimates and the Dirac Operator. In addition, the
paper proves the existence of Riemannian manifolds with arbitrarily bounded scalar
curvature and arbitrary volume.
Work in Progress
- Geometric Effect of Contracting Mappings onto Spheres (ART project)
- Pascal-like Triangles as Matrices (ART project)
- Counting Finite Topologies (CFR project)
|
|
