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Search: MSC category 53C20 ( Global Riemannian geometry, including pinching [See also 31C12, 58B20] )

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1. CMB 2011 (vol 55 pp. 632)

Pigola, S.; Rimoldi, M.
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

2. CMB 2010 (vol 53 pp. 684)

Proctor, Emily; Stanhope, Elizabeth
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

3. CMB 2010 (vol 53 pp. 412)

Calvaruso, G.
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

4. CMB 2007 (vol 50 pp. 24)

5. CMB 2006 (vol 49 pp. 321)

Balser, Andreas
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

6. CMB 2006 (vol 49 pp. 226)

Engman, Martin
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

7. CMB 2006 (vol 49 pp. 152)

Yun, Jong-Gug
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

8. CMB 2004 (vol 47 pp. 314)

Yun, Jong-Gug
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

9. CMB 2003 (vol 46 pp. 617)

Pak, Hong Kyung
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

10. CMB 2003 (vol 46 pp. 130)

Petersen, Peter; Wilhelm, Frederick
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

11. CMB 2000 (vol 43 pp. 343)

Hughes, Bruce; Taylor, Larry; Williams, Bruce
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

12. CMB 2000 (vol 43 pp. 74)

Kimura, Makoto; Maeda, Sadahiro
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

13. CMB 1999 (vol 42 pp. 214)

Paeng, Seong-Hun; Yun, Jong-Gug
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

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