Sectional Curvature in Piecewise Linear Manifolds

A metric complex M is a connected, locally-finite simplicial complex linearly embedded in some Euclidean space R. Metric complexes M and M' are isometric if they have subdivisions L and L and if there is a simplicial isomorphism h:L -• L such that for every a e L, the linear map determined by h\a -• h(a) is an isometry (that is, it extends to an isometry of the affine spaces generated by these simplexes). This note is concerned with certain characteristics of a metric complex M which are intrinsic, i.e., which depend only on the isometry class of M. The basic such characteristic is the intrinsic metric, which is best described in the piecewise linear context by H. Gluck [3]; for a more general treatment see W. Rinow [8]. Let M ç R be a metric complex and let p be a point of M. Then the tangent cone TPM of M at p is defined to be the infinite cone with vertex p generated by link(/?, M). The isometry class of TPM is intrinsic to M, for each p. An infinite ray px in TpM will be called a tangent direction at p to M. Let NPM be a subcone of TPM and let j be a nonnegative integer. Let R x NPM be given the metric in which its factors are orthogonal. For various choices of NpM and j \ R j x NPM will be isometric to TPM. For example if p is in the interior of a /-simplex of M, such a factoring exists. Consider those factorings of TPM for which j is maximal; then the corresponding NPM are all isometric. Such an NPM will be called the normal geometry of p in M, and denoted vpM. For example, if M is an «-manifold and p is in the interior of an (n — 1)or «-simplex, then vpM = {/?}. If M is a 2-manifold, then vpM = {p} unless/? is a vertex of nonzero curvature, when vpM — TPM. Clearly j and vpM determine the metric geometry of M near p. For any pe M and any tangent direction px at p lying in vpM I have defined numbers k+(px) and k_(px), with k+(px) ^ k-(px), called the maximum and minimum curvatures of M at /? in the direction /?3c. The definitions are too long to give here. Roughly speaking, k-(px) equals: 2n minus twice the maximum "angle" that can occur between px and any other py £ vpM as y varies; k+{px) is defined similarly, using a