Polyhedral space is a certain

metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general sett ...

. A ( Euclidean) polyhedral space is a (usually finite)

simplicial complex In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial ...

in which every

simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension ...

has a flat metric. (Other spaces of interest are spherical and hyperbolic polyhedral spaces, where every simplex has a metric of constant positive or negative curvature). In the sequel all polyhedral spaces are taken to be Euclidean polyhedral spaces.

Examples

All 1-dimensional polyhedral spaces are just

metric graph In mathematics and physics, a quantum graph is a linear, network-shaped structure of vertices connected on edges (i.e., a graph) in which each edge is given a length and where a differential (or pseudo-differential) equation is posed on each edge ...

s. A good source of 2-dimensional examples constitute triangulations of 2-dimensional surfaces. The surface of a convex polyhedron in

R^3

is a 2-dimensional polyhedral space. Any PL-manifold (which is essentially the same as a simplicial manifold, just with some technical assumptions for convenience) is an example of a polyhedral space. In fact, one can consider pseudomanifolds, although it makes more sense to restrict the attention to normal manifolds.

Metric singularities

In the study of polyhedral spaces (particularly of those that are also

topological manifold In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real ''n''- dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout ma ...

s) metric singularities play a central role. Let a polyhedral space be an n-dimensional manifold. If a point in a polyhedral space that is an n-dimensional topological manifold has no neighborhood isometric to a Euclidean neighborhood in R^n, this point is said to be a metric singularity. It is a singularity of codimension k, if it has a neighborhood isometric to R^ with a

metric cone Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathem ...

. Singularities of codimension 2 are of major importance; they are characterized by a single number, the conical angle. The singularities can also studied topologically. Then, for example, there are no topological singularities of codimension 2. In a 3-dimensional polyhedral space without a boundary (faces not glued to other faces) any point has a neighborhood homeomorphic either to an open ball or to a cone over the

projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that ...

. In the former case, the point is necessarily a codimension 3 metric singularity. The general problem of topologically classifying singularities in polyhedral spaces is largely unresolved (apart from simple statements that e.g. any singularity is locally a cone over a spherical polyhedral space one dimension less and we can study singularities there).

Curvature

It is interesting to study the curvature of polyhedral spaces (the curvature in the sense of Alexandrov spaces), specifically polyhedral spaces of nonnegative and nonpositive curvature. Nonnegative curvature on singularities of codimension 2 implies nonnegative curvature overall. However, this is false for nonpositive curvature. For example, consider R^3 with one octant removed. Then on the edges of this octant (singularities of codimension 2) the curvature is nonpositive (because of branching geodesics), yet it is not the case at the origin (singularity of codimension 3), where a triangle such as (0,0,e), (0,e,0), (e,0,0) has a median longer than would be in the Euclidean plane, which is characteristic of nonnegative curvature.

Additional structure

Many concepts of

Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to po ...

can be applied. There is only one obvious notion of

parallel transport In geometry, parallel transport (or parallel translation) is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection on the tangent b ...

and only one natural connection. The concept of

holonomy In differential geometry, the holonomy of a connection on a smooth manifold is a general geometrical consequence of the curvature of the connection measuring the extent to which parallel transport around closed loops fails to preserve the geome ...

is strikingly simple in this case. The

restricted holonomy group Restricted may refer to: *R rating (disambiguation), list of subjects where "R" stands for "Restricted" *18 rating, media rating designation sometimes called "Restricted" *Restricted (country club), historical use of the term in country clubs in t ...

is trivial, and so there is a

homomorphism In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "sa ...

from the

fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...

onto the holonomy group. It may be especially convenient to remove all singularities to obtain a space with a flat Riemannian metric and to study the holonomies there. One concepts thus arising are polyhedral Kähler manifolds, when the holonomies are contained in a group, conjugate to the

unitary matrices In linear algebra, a complex square matrix is unitary if its conjugate transpose is also its inverse, that is, if U^* U = UU^* = UU^ = I, where is the identity matrix. In physics, especially in quantum mechanics, the conjugate transpose is ...

. In this case, the holonomies also preserve a

symplectic form In mathematics, a symplectic vector space is a vector space ''V'' over a field ''F'' (for example the real numbers R) equipped with a symplectic bilinear form. A symplectic bilinear form is a mapping that is ; Bilinear: Linear in each argument ...

, together with a complex structure on this polyhedral space (manifold) with the singularities removed. All the concepts such as

differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many application ...

, L2 differential form, etc. are adjusted accordingly.

History

In full generality, polyhedral spaces were first defined by Milka Milka, A. D. Multidimensional spaces with polyhedral metric of nonnegative curvature. I. (Russian) Ukrain. Geometr. Sb. Vyp. 5--6 1968 103–114.

References