Bieberbach groups and flat manifolds with finite abelian holonomy from Artin braid groups

Let [Formula: see text]. In this paper, we show that for any abelian subgroup [Formula: see text] of [Formula: see text] the crystallographic group [Formula: see text] has Bieberbach subgroups [Formula: see text] with holonomy group [Formula: see text]. Using this approach, we obtain an explicit description of the holonomy representation of the Bieberbach group [Formula: see text]. As an application, when the holonomy group is cyclic of odd order, we study the holonomy representation of [Formula: see text] and determine the existence of Anosov diffeomorphisms and Kähler geometry of the flat manifold [Formula: see text] with fundamental group the Bieberbach group [Formula: see text].

Profinite genus of the fundamental groups of compact flat manifolds with holonomy group of prime order

In this article, we calculate the profinite genus of the fundamental group of an 𝑛-dimensional compact flat manifold 𝑋 with holonomy group of prime order. As consequence, we prove that if n ⩽ 21 n\leqslant 21 , then 𝑋 is determined among all 𝑛-dimensional compact flat manifolds by the profinite completion of its fundamental group. Furthermore, we characterize the isomorphism class of the profinite completion of the fundamental group of 𝑋 in terms of the representation genus of its holonomy group.

Hypersurfaces in a conformally flat space

The hypersurface of a space is one of the most important objects in a space. Many authors studied the various geometric aspects of hypersurfaces in a space form. The notion of conformal flatness is one of the most primitive concepts in differential geometry. Again, conformally flat space is a proper generalization of a space form. In this paper, we study the geometry of hypersurfaces in a conformally flat manifold. Then we have investigated some sufficient conditions imposed on the shape operator for which the hypersurface satisfies various pseudosymmetric-type conditions imposed on its conformal curvature tensor.

New Results About the Lambda Constant and Ground States of the 𝑊-Functional

In this paper, we study properties of the lambda constants and the existence of ground states of Perelman’s famous W-functional from a variational formulation. We have two kinds of results. One is about the estimation of the lambda constant of G. Perelman, and the other is about the existence of ground states of his W-functional, both on a complete non-compact Riemannian manifold {(M,g)}. One consequence of our estimation is that, on an ALE (or asymptotic flat) manifold {(M,g)}, if the scalar curvature s of {(M,g)} is non-negative and has quadratical decay at infinity, then M is scalar flat, i.e., {s=0} in M. We also introduce a new constant {d(M,g)}. For the existence of the ground states, we use Lions’ concentration-compactness method.

Brownian motion on Perelman’s almost Ricci-flat manifold

We study Brownian motion and stochastic parallel transport on Perelman’s almost Ricci flat manifold {\mathcal{M}=M\times\mathbb{S}^{N}\times I}, whose dimension depends on a parameter N unbounded from above. We construct sequences of projected Brownian motions and stochastic parallel transports which for {N\to\infty} converge to the corresponding objects for the Ricci flow. In order to make precise this process of passing to the limit, we study the martingale problems for the Laplace operator on {\mathcal{M}} and for the horizontal Laplacian on the orthonormal frame bundle {\mathcal{OM}}. As an application, we see how the characterizations of two-sided bounds on the Ricci curvature established by A. Naber applied to Perelman’s manifold lead to the inequalities that characterize solutions of the Ricci flow discovered by Naber and the second author.

Intrinsic flat stability of the positive mass theorem for graphical hypersurfaces of Euclidean space

The rigidity of the Positive Mass Theorem states that the only complete asymptotically flat manifold of nonnegative scalar curvature and

CONFORMALLY FLAT CONTACT THREE-MANIFOLDS

In this paper, we consider contact metric three-manifolds $(M;\unicode[STIX]{x1D702},g,\unicode[STIX]{x1D711},\unicode[STIX]{x1D709})$ which satisfy the condition $\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x1D709}}h=\unicode[STIX]{x1D707}h\unicode[STIX]{x1D711}+\unicode[STIX]{x1D708}h$ for some smooth functions $\unicode[STIX]{x1D707}$ and $\unicode[STIX]{x1D708}$, where $2h=\unicode[STIX]{x00A3}_{\unicode[STIX]{x1D709}}\unicode[STIX]{x1D711}$. We prove that if $M$ is conformally flat and if $\unicode[STIX]{x1D707}$ is constant, then $M$ is either a flat manifold or a Sasakian manifold of constant curvature $+1$. We cannot extend this result for a smooth function $\unicode[STIX]{x1D707}$. Indeed, we give such an example of a conformally flat contact three-manifold which is not of constant curvature.