# nLab Alexandrov space

Alexandrov spaces

This entry is about length spaces behaving like having sectional curvature bounded below, named after Alexander Alexandrov. For specialization topologies on preorders see instead at Alexandroff spaces, named after Paul Alexandroff.

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

infinitesimal cohesion

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

#### Riemannian geometry

Riemannian geometry

# Alexandrov spaces

## Idea

In differential geometry, Alexandrov spaces are certain locally compact complete length spaces which generalize Riemannian manifolds whose sectional curvature is bounded below by some real number.

## References

• Katsuhiro Shiohama, An Introduction to the Geometry of Alexandrov spaces, lecture at Daewoo Workshop on Differential Geometry (1992) [pdf]

• Yukio Otsu, Differential geometric aspects of Alexandrov spaces, in: Comparison Geometry, MSRI publications 30, Cambridge University Press (1997) [pdf, ISBN-10: 0-521-59222-4]

• Wikipedia, Alexandrov space

• Grigori Perelman, Elements of Morse theory on Alexandrov spaces, St. Petersburg Math. J. 5 1 (1994) 205–213 [mathet:aa374]

• Grigori Perelman, DC structure on Alexandrov spaces [pdf, pdf]

• Yuri Burago, Mikhail Gromov, Grigori Perelman, Alexandrov spaces with curvature bounded below, Russ. Math. Surveys 41 3 (1992) 1-54 [doi:10.1070/RM1992v047n02ABEH000877, pdf]

• MathOverflow, Details of Perelman’s example about soul of Alexandrov space (2010) [MO:q/22105]

• MathOverflow, Metrically singular Alexandrov space (2011) [MO:q/67046],

• MathOverflow, Is there Domain Invariance for Alexandrov spaces? (2010) [MO:q/21512]

• MathOverflow, Rigidity of triangle comparison in Alexandrov spaces (2010) [MO:q/44943]

Last revised on June 2, 2024 at 05:58:30. See the history of this page for a list of all contributions to it.