## Geodesics in piecewise linear manifolds

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- by David A. Stone PDF
- Trans. Amer. Math. Soc.
**215**(1976), 1-44 Request permission

## Abstract:

A simplicial complex**M**is

*metrized*by assigning to each simplex $a \in {\mathbf {M}}$ a linear simplex ${a^\ast }$ in some Euclidean space ${{\mathbf {R}}^k}$ so that face relations correspond to isometries. An equivalence class of metrized complexes under the relation generated by subdivisions and isometries is called a

*metric complex*; it consists primarily of a polyhedron

**M**with an intrinsic metric ${\rho _{\mathbf {M}}}$. This paper studies geodesics in metric complexes. Let $P \in {\mathbf {M}}$; then the tangent space ${T_P}({\mathbf {M}})$ is canonically isometric to an orthogonal product of cones from $P,{{\mathbf {R}}^k} \times {\nu _P}({\mathbf {M}})$; once

*k*is as large as possible. ${\nu _P}({\mathbf {M}})$ is called the

*normal geometry*at

*P*in

**M**. Let $P\bar X$ be a tangent direction at

*P*in ${\nu _P}({\mathbf {M}})$. I define numbers ${\kappa _ + }(P\bar X)$ and ${\kappa _ - }(P\bar X)$, called the

*maximum*and

*minimum curvatures*at

*P*in the direction $P\bar X$. THEOREM.

*Let*

**M**

*be a complete, simply-connected metric complex which is a p.l. n-manifold without boundary. Assume*${\kappa _ + }(P\bar X) \leqslant 0$

*for all*$P \in {\mathbf {M}}$

*and all*$P\bar X \subseteq {\nu _P}({\mathbf {M}})$.

*Then*

**M**

*is p.l. isomorphic to*${{\mathbf {R}}^n}$. This is analogous to a well-known theorem for smooth manifolds by E. Cartan and J. Hadamard. THEOREM (ROUGHLY).

*Let*

**M**

*be a complete metric complex which is a p.l. n-manifold without boundary. Assume*(1)

*there is a number*$\kappa \geqslant 0$

*such that*${\kappa _ - }(P\bar X) \geqslant \kappa$

*whenever P is in the*$(n - 2)$-

*skeleton of*

**M**

*and whenever*$P\bar X \subseteq {\nu _P}({\mathbf {M}})$; (2)

*the simplexes of*

**M**

*are bounded in size and shape. Then*

**M**

*is compact*. This is analogous to a weak form of a well-known theorem of S. B. Myers for smooth manifolds.

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## Additional Information

- © Copyright 1976 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**215**(1976), 1-44 - MSC: Primary 53C20; Secondary 57C25
- DOI: https://doi.org/10.1090/S0002-9947-1976-0402648-8
- MathSciNet review: 0402648