Expand all Collapse all | Results 1 - 14 of 14 |
1. CMB Online first
On the maximum curvature of closed curves in negatively curved manifolds Motivated by Almgren's work on the isoperimetric inequality,
we prove a sharp inequality relating the length and maximum curvature
of a closed curve in a complete, simply connected manifold of
sectional curvature at most $-1$. Moreover, if equality holds,
then the norm of the geodesic curvature is constant and the torsion
vanishes. The proof involves an application of the maximum principle
to a function defined on pairs of points.
Keywords:manifold, curvature Category:53C20 |
2. CMB 2011 (vol 55 pp. 632)
Characterizations of Model Manifolds by Means of Certain Differential Systems We prove metric rigidity for complete manifolds supporting solutions of
certain second order differential systems, thus extending classical works on a
characterization of space-forms. Along the way, we also discover
new characterizations of space-forms. We next generalize results concerning metric
rigidity via equations involving vector fields.
Keywords:metric rigidity, model manifolds, Obata's type theorems Category:53C20 |
3. CMB 2010 (vol 53 pp. 684)
An Isospectral Deformation on an Infranil-Orbifold
We construct a Laplace isospectral deformation of metrics on an
orbifold quotient of a nilmanifold. Each orbifold in the deformation
contains singular points with order two isotropy. Isospectrality is
obtained by modifying a generalization of Sunada's theorem due to
DeTurck and Gordon.
Keywords:spectral geometry, global Riemannian geometry, orbifold, nilmanifold Categories:58J53, 53C20 |
4. CMB 2010 (vol 53 pp. 412)
Einstein-Like Lorentz Metrics and Three-Dimensional Curvature Homogeneity of Order One We completely classify three-dimensional Lorentz manifolds, curvature homogeneous up to order one, equipped with Einstein-like metrics. New examples arise with respect to both homogeneous examples and three-dimensional Lorentz manifolds admitting a degenerate parallel null line field.
Keywords:Lorentz manifolds, curvature homogeneity, Einstein-like metrics Categories:53C50, 53C20, 53C30 |
5. CMB 2007 (vol 50 pp. 24)
Invariant Metrics with Nonnegative Curvature on Compact Lie Groups We classify the left-invariant metrics with nonnegative sectional curvature on $\SO(3)$ and $U(2)$.
Category:53C20 |
6. CMB 2006 (vol 49 pp. 321)
Polygons with Prescribed Gauss Map in Hadamard Spaces and Euclidean Buildings We show that given a stable weighted configuration on the asymptotic
boundary of a
locally compact Hadamard space, there is a polygon with Gauss
map prescribed by the given weighted configuration.
Moreover, the same result holds for
semistable configurations on arbitrary Euclidean buildings.
Keywords:Euclidean buildings, Hadamard spaces, polygons Category:53C20 |
7. CMB 2006 (vol 49 pp. 226)
The Spectrum and Isometric Embeddings of Surfaces of Revolution A sharp upper bound on the first $S^{1}$ invariant eigenvalue of the Laplacian
for $S^1$ invariant metrics on $S^2$ is used to find obstructions to the existence
of $S^1$ equivariant isometric embeddings of such metrics in $(\R^3,\can)$. As a
corollary we prove: If the first four distinct eigenvalues have even multiplicities
then the metric cannot be equivariantly, isometrically embedded in $(\R^3,\can)$. This
leads to generalizations of some classical results in the theory of surfaces.
Categories:58J50, 58J53, 53C20, 35P15 |
8. CMB 2006 (vol 49 pp. 152)
Comparison Geometry With\\$L^1$-Norms of Ricci Curvature We investigate the geometry of manifolds with bounded Ricci
curvature in $L^1$-sense. In particular, we generalize the
classical volume comparison theorem to our situation and obtain a
generalized sphere theorem.
Keywords:Mean curvature, Ricci curvature Category:53C20 |
9. CMB 2004 (vol 47 pp. 314)
Mean Curvature Comparison with $L^1$-norms of Ricci Curvature We prove an analogue of mean curvature comparison theorem in the case where the
Ricci curvature below a positive constant is small in $L^1$-norm.
Keywords:mean curvature, Ricci curvature Category:53C20 |
10. CMB 2003 (vol 46 pp. 617)
On Harmonic Theory in Flows Recently [8], a harmonic theory was developed for a compact
contact manifold from the viewpoint of the transversal geometry of
contact flow. A contact flow is a typical example of geodesible
flow. As a natural generalization of the contact flow, the present
paper develops a harmonic theory for various flows on compact
manifolds. We introduce the notions of $H$-harmonic and
$H^*$-harmonic spaces associated to a H\"ormander flow. We also
introduce the notions of basic harmonic spaces associated to a weak
basic flow. One of our main results is to show that in the special
case of isometric flow these harmonic spaces are isomorphic to the
cohomology spaces of certain complexes. Moreover, we find an
obstruction for a geodesible flow to be isometric.
Keywords:contact structure, geodesible flow, isometric flow, basic cohomology Categories:53C20, 57R30 |
11. CMB 2003 (vol 46 pp. 130)
On Frankel's Theorem In this paper we show that two minimal hypersurfaces in a manifold with
positive Ricci curvature must intersect. This is then generalized to show
that in manifolds with positive Ricci curvature in the integral sense two
minimal hypersurfaces must be close to each other. We also show
what happens if a manifold with nonnegative Ricci curvature admits
two nonintersecting minimal hypersurfaces.
Keywords:Frankel's Theorem Category:53C20 |
12. CMB 2000 (vol 43 pp. 343)
Controlled Homeomorphisms Over Nonpositively Curved Manifolds We obtain a homotopy splitting of the forget control map for
controlled homeomorphisms over closed manifolds of nonpositive
curvature.
Keywords:controlled topology, controlled homeomorphism, nonpositive curvature, Novikov conjectures Categories:57N15, 53C20, 55R65, 57N37 |
13. CMB 2000 (vol 43 pp. 74)
Geometric Meaning of Isoparametric Hypersurfaces in a Real Space Form We shall provide a characterization of all isoparametric hypersurfaces
$M$'s in a real space form $\tilde{M}(c)$ by observing the extrinsic
Wshape of geodesics of $M$ in the ambient manifold $\tilde{M}(c)$.
Categories:53C35, 53C20, 53C22 |
14. CMB 1999 (vol 42 pp. 214)
Conjugate Radius and Sphere Theorem Bessa [Be] proved that for given $n$ and $i_0$, there exists
an $\varepsilon(n,i_0)>0$ depending on $n,i_0$ such that if $M$
admits a metric $g$ satisfying $\Ric_{(M,g)} \ge n-1$, $\inj_{(M,g)}
\ge i_0>0$ and $\diam_{(M,g)} \ge \pi-\varepsilon$, then $M$ is
diffeomorphic to the standard sphere. In this note, we improve this
result by replacing a lower bound on the injectivity radius with a
lower bound of the conjugate radius.
Keywords:Ricci curvature, conjugate radius Categories:53C20, 53C21 |