Gauss curvatures and tube for polyhedron
by
Jin-ichi Itoh
Kumamoto University, Japan

Oral Communication

We define some curvatures of polyhedron which corresponds to Gauss curvature and show Gauss's Theorema Egregium, Chern-Lashof's inequality and Weyl's volume formula of tube for polyhedron.

When a d dimensional combinatorial manifold P in R^d+1 satisfies the following (i) and (ii), we call it a d dimensional polyhedral manifold. (i) For any m dimensional simplex \sigma^m has a flat metric such that \sigma^m coincides with a linear simplex in R^m+1. (ii) For any vertex v, locally embedded around each vertex. We say a vertex v with it property (*), if there is a point n in S^d such that for any normal vecor n_i of face around v \angle(n_i, n) < \frac\pi2,

We define the intrinsic curvature Kⁱ(v) at a vertex v in P by

Ki(v): = n å m=0  (-1)m å \sigmajm \owns v  \betajm,

where \beta^m is the outer angle of m-simplex at v. Next we will define the extrinsic curvature only 2-dimensional case. We denote the faces of P around v by {f_i}  (i=1, ... , k), its unit normal vectors of one side by {n_i}, where the order of indices i is anti-clockwise looking from the side of normal directions. Let c_{j, j+1} be a great arc from n_j to n_j+1 on S². We define the closed brocken great arcs c by c: = c_{1, 2} \cup c_{2, 3} \cup ... \cup c_{k, 1} and define K^e(v, n) by the signed area of enclosed domain by c with respect to a base point n as follows. We denote the oriented spherical triangle with vertices n,   n_j,   n_j+1 by T(j). Let J⁺ (resp. J^-) be the subset of {1, ... , k} such that the orientation of T(j) is anti-clockwise (resp. clockwise). We define K^e(v, n) by

Ke(v, n) : = \frac12\pi( å j in J+  Area(T(j)) - å j in J-  Area(T(j)))

When a polyhedron P has property (*), the value of K^e(v, n) does not depend on the choice of n, then we write it K^e(v) and call it the extrinsic curvature of v. We omit the definition of general dimension. Note that the above definition is different from Banchoff's one.

Theorem(Theorema egregium for polyhedron) For any d-dimensional polyhedron P in R^d+1 and vertex v in P with property (*) it holds that Kⁱ(v) = K^e(v).

It is well known that Gauss-Bonnet type equality holds (i.e. \sum_{v in P} Kⁱ(v) = \chi(P) , where \chi(P) denote the Euler number of P.) We consider the total absolute curvature for a 2-dimensional polyhedron. We denote the domains devived by the boundaries of spherical triangle T(j) by D(i). For each domain D(i) define the number m_i by m_i ^+/- : = # {j in J ^+/- | D(i) subset T(j) },  m_i : = m_i⁺ - m_i^-. Define the absolute curvature K^a(v) at v by

Ka(v) : = \frac12\pi å i  |mi| Area(D(i)).

Theorem (Chern-Lashoff type ineq. for polyhedron) For any d-dimensional polyhedron P in R^d+1 satisfying property (*) at all vertices it holds that

å v in P  Ka(v) >= b(P),

where b denotes the sum of all dimensional Betti numbers.

The Weyl's volume formula of tube for a closed surface M in R³ says that for enough small positive number r, Vol V_r(M) = 2r Area M + \frac2r³3 \int_M K dv, where V_r(M) denotes the r neighborhood of M, K denotes Gauss curvature on M. Let P be a 2-dimensional polyhedron in R³ . We assume for simplicity that the number of face around v is three. We define some kind of curvature [K\tilde](v) at v as follows. Let T_j   (j=1, 2, 3) be the tangent plane of S² at n_j, and T_j⁺ (resp. T_j^-) be the half space with boundary T_j and containing (resp. not containing) the origin. Let H_j be the plane through the origin and two vertices except n_j , and H_j⁺ (resp. H_j^- ) be the half space with boundary H_j and not containing (resp. containing) n_j. Put q: = \cap _j=1ⁿ T_j. When q in \cap _{j=1, 2, 3}H_j⁺, define [K\tilde](v) by the volume of \cap _{j=1, 2, 3}H_j⁺ \cap \cap _{j=1, 2, 3}T_j⁺. When q in H_j^- \cap H_j+1⁺ \cap H_j+2⁺, define [K\tilde](v) by

Vol( Ç j=1, 2, 3  Hj+ \cap Ç j=1, 2, 3  Tj+)- Vol(Hj- \cap Hj+1+ \cap Hj+2+ \cap Tj- \cap Tj+1+ \cap Tj+2+).

Theorem (Weyl type volume formula of tube for polyhedron) For any polyhedron P in R³ and enough small positive number r, it holds that

Vol Vr(P) = 2r Area(P) + \frac2r33 å v in P  ~ K   (v).

Under an adequate definition of K_i for d dimensional closed polyhedron P in R^d+1 it holds that if d is odd (resp. even),

Vol Vr(P) = 2{ r Area(P) + \fracr33 Kd-3(P) + ... +\fracrdd K0(P)   (resp.   \fracrd-1d-1 K1(P) ) }.

Date received: May 27, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadq-68.