CAT(k) space and Orbifold

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Difference between CAT(k) space and Orbifold

CAT(k) space vs. Orbifold

In mathematics, a \mathbf space, where k is a real number, is a specific type of metric space. In the mathematical disciplines of topology, geometry, and geometric group theory, an orbifold (for "orbit-manifold") is a generalization of a manifold.

Contractible space

In mathematics, a topological space X is contractible if the identity map on X is null-homotopic, i.e. if it is homotopic to some constant map.

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Euclidean space

In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.

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Homotopy

In topology, two continuous functions from one topological space to another are called homotopic (from Greek ὁμός homós "same, similar" and τόπος tópos "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions.

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Hyperbolic space

In mathematics, hyperbolic space is a homogeneous space that has a constant negative curvature, where in this case the curvature is the sectional curvature.

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Metric space

In mathematics, a metric space is a set for which distances between all members of the set are defined.

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Mikhail Leonidovich Gromov

Mikhail Leonidovich Gromov (also Mikhael Gromov, Michael Gromov or Mischa Gromov; Михаи́л Леони́дович Гро́мов; born 23 December 1943), is a French-Russian mathematician known for work in geometry, analysis and group theory.

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Non-positive curvature

In mathematics, spaces of non-positive curvature occur in many contexts and form a generalization of hyperbolic geometry.

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Riemannian manifold

In differential geometry, a (smooth) Riemannian manifold or (smooth) Riemannian space (M,g) is a real, smooth manifold M equipped with an inner product g_p on the tangent space T_pM at each point p that varies smoothly from point to point in the sense that if X and Y are differentiable vector fields on M, then p \mapsto g_p(X(p),Y(p)) is a smooth function.

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Simply connected space

In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the space) into any other such path while preserving the two endpoints in question.

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Trivial group

In mathematics, a trivial group is a group consisting of a single element.

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