Almost flat manifold - Misplaced Pages

(Redirected from Gromov-Ruh theorem)

In mathematics, a smooth compact manifold M is called almost flat if for any $\varepsilon >0$ there is a Riemannian metric $g_{\varepsilon }$ on M such that ${\mbox{diam}}(M,g_{\varepsilon })\leq 1$ and $g_{\varepsilon }$ is $\varepsilon$ -flat, i.e. for the sectional curvature of $K_{g_{\varepsilon }}$ we have $|K_{g_{\epsilon }}|<\varepsilon$ .

Given $n$ , there is a positive number $\varepsilon _{n}>0$ such that if an $n$ -dimensional manifold admits an $\varepsilon _{n}$ -flat metric with diameter $\leq 1$ then it is almost flat. On the other hand, one can fix the bound of sectional curvature and get the diameter going to zero, so the almost-flat manifold is a special case of a collapsing manifold, which is collapsing along all directions.

According to the Gromov–Ruh theorem, $M$ is almost flat if and only if it is infranil. In particular, it is a finite factor of a nilmanifold, which is the total space of a principal torus bundle over a principal torus bundle over a torus.

References

Hermann Karcher. Report on M. Gromov's almost flat manifolds. Séminaire Bourbaki (1978/79), Exp. No. 526, pp. 21–35, Lecture Notes in Math., 770, Springer, Berlin, 1980.
Peter Buser and Hermann Karcher. Gromov's almost flat manifolds. Astérisque, 81. Société Mathématique de France, Paris, 1981. 148 pp.
Peter Buser and Hermann Karcher. The Bieberbach case in Gromov's almost flat manifold theorem. Global differential geometry and global analysis (Berlin, 1979), pp. 82–93, Lecture Notes in Math., 838, Springer, Berlin-New York, 1981.
Gromov, M. (1978), "Almost flat manifolds", Journal of Differential Geometry, 13 (2): 231–241, doi:10.4310/jdg/1214434488, MR 0540942.
Ruh, Ernst A. (1982), "Almost flat manifolds", Journal of Differential Geometry, 17 (1): 1–14, doi:10.4310/jdg/1214436698, MR 0658470.

v t e Riemannian geometry (Glossary)
Basic concepts	Curvature tensor Scalar Ricci Sectional Exponential map Geodesic Inner product Metric tensor Levi-Civita connection Covariant derivative Signature Raising and lowering indices/Musical isomorphism Parallel transport Riemannian manifold Pseudo-Riemannian manifold Riemannian volume form
Types of manifolds	Hermitian Hyperbolic Kähler Kenmotsu
Main results	Fundamental theorem of Riemannian geometry Gauss's lemma Gauss–Bonnet theorem Hopf–Rinow theorem Nash embedding theorem Ricci flow Schur's lemma
Generalizations	Finsler Hilbert Sub-Riemannian
Applications	General relativity Geometrization conjecture Poincaré conjecture Uniformization theorem

Manifolds (Glossary, List, Category)

Basic concepts

Main results (list)

Maps

Types of
manifolds

Tensors

Vectors	Distribution Lie bracket Pushforward Tangent space bundle Torsion Vector field Vector flow
Covectors	Closed/Exact Covariant derivative Cotangent space bundle De Rham cohomology Differential form Vector-valued Exterior derivative Interior product Pullback Ricci curvature flow Riemann curvature tensor Tensor field density Volume form Wedge product
Bundles	Adjoint Affine Associated Cotangent Dual Fiber (Co) Fibration Jet Lie algebra (Stable) Normal Principal Spinor Subbundle Tangent Tensor Vector
Connections	Affine Cartan Ehresmann Form Generalized Koszul Levi-Civita Principal Vector Parallel transport

Generalizations

This Riemannian geometry-related article is a stub. You can help Misplaced Pages by expanding it.

Categories: