数理解析研究所講究録
第 1562 巻 2007 年 87-91

87

The Gauss-Bonnet theorem for PL manifolds\sim Banchoff’s theorem and Homma’s theorem\simSAT\^O, Kenzi
佐藤健治

There is the Gauss-Bonnet theorem not only for smooth surfaces but also for polyhedra. It can begeneralized to the $th\infty rem$ for higher dimensional PL mfds, in two methods of Banchoff’s and Homma’s.
The purpose of this article is to cooider the relation of them. First of all, the case of dimension 2 has tobe described.

.

Gauss’ Theorema egregium.
For the boundaryof a convex polyhedrvs $P=P^{3}$ of$2-dim$$\partial P$$R^{3}$, let$\sum^{m-1}(\pi-\theta:)-(m-2)\pi$(G)$\kappa(\langle v\})=2\cdot\frac{:-0}{4\pi}d\epsilon f$$(= \lim\frac{2}{4\pi}\int_{\epsilon ma1ln.b.d..tvof\delta P}KdV)$.

Then(B)$\kappa(\{v\})=1-\frac{1}{2}\cdot m+\sum_{:=0}^{m-1}\frac{\pi-\theta_{1}}{2\pi}$$=1- \sum_{:=0}^{m-1}\frac{\theta_{1}}{2\pi}$.

(H)

See the figure below.

$\ell_{:}\perp l_{j}$The value$(B)=(II)$.

$\kappa(\{v\})$depends only$\partial P$

if $t\neq j$, so, for a general PL-mfd $M=M^{2}$ , we can define the curvature$\kappa$byThis is an abetract and the details witl be published elsewhere. The title in Japanme is “PL 多様体の $G-$-Bonnetの定理 $\sim Banchoff$ の定理と本間の定理\sim ’|.

88.

Gauss-Bonnet theorem.
We have $\sum_{v\in M}\kappa(\{v\})=\chi(M)$ .

$2-dim$$\frac{1}{4}\cdot 8=2=\chi(S^{2})$$7^{1}124=2=\chi(S^{2})$$(-18) \cdot 12+\frac{1}{6}\cdot 12=0=\chi(T^{2})$To generalize them for the higher dimension, the inner and outer angles for vertices are confirmed by thefollowing figure.
$\alpha(\{v\},Q’)=\frac{\theta}{2\pi}$$\alpha^{o}(\{v\},Q’)=\frac{\theta^{O}}{2\pi}$$\alpha(\{v\},Q)=\frac{S}{4\pi}$$\alpha^{o}(\{v\},Q)=\frac{s\circ}{4\pi}$As follows, Theorema egregium and the Gauss-Bonnet thorem are extended.
Gauss’ Theorema egregium.
For the boundaryof a convex polyhedra $P=P^{n}$ of$\partial P$$R^{n}$, let(G)$\kappa_{B}(\{v\})=dof(1+(-1)^{nrightarrow 1})\alpha^{o}(\{v\}, P)$$(=$ $\lim\frac{1+(-1)^{n-1}}{\omega_{n-1}}\int_{-ma1ln.b.d.atv}$
of$\partial PKdV$).

Then$\kappa_{B}(\{v\})=\sum_{Q\sigma\partial P}^{v\epsilon 0}(-1)^{|Q|}\alpha^{o}(\{v\},Q)$.

So, for a general PL-mfd $M=M^{n-1}$ , let$\kappa_{B}(\{v\})d=\sum_{Q\subseteqq u}^{v\epsilon Q}(-1)^{|Q|}\alpha^{o}(\{v\},Q)of$(B)89
Gauss-Bonnet theorem (Banchoff).
$\sum_{v\in M}\kappa_{B}(\{v\})=\chi(M)$.

We have another generalization.

Gauss-Bonnet theorem (Homma).
For ench face $Q’$ of $M$ , let$\kappa_{H}(Q’)d=ef1-\sum_{Q\subseteqq M}^{Q’\subseteqq Q}\alpha(Q’,Q)|Q|=n-1$

(H)

Then
$\sum_{Q\subseteqq M}(-1)^{|Q’|}\kappa_{H}(Q’)=\chi(M)$.

We also have to confirm the definition of angles for faces. See the figure below.
.
$\cdot$.

.

$\cdot$$\alpha(\ell,Q)=\frac{\theta}{2\pi}$$\alpha^{O}(\ell,Q)=\frac{\theta^{o}}{2\pi}$Notice that $\alpha(Q’, Q)=\alpha^{O}(Q’, Q)=51$ if $|Q|-|Q’|=1$ , and $\alpha(Q, Q)=\alpha^{O}(Q, Q)=1$ . The following is
the main result of this article.

Relation of Banchoff and Homma (S.).
For ench $v\in M$ , we have$\kappa_{B}(\{v\})=\sum_{Q_{=}^{C}M}^{v\in Q}(-1)^{|Q|}\kappa_{H}(Q)\cdot\alpha^{O}(\{v\},Q)$

Remark that$\sum_{v\in Q}\alpha^{O}(\{v\},Q)=1$

for$\forall_{Q}\subseteqq M$.

.

$\alpha(\{v\},Q)=\frac{\theta}{2\pi}$$\alpha^{o}(\{v\},Q)=\frac{\theta^{o}}{2\pi}$90
Extendfor all faces (and extend$\kappa_{B}$$\kappa_{B}$and$\kappa_{H}$for $P=P^{n}$ ) by$\kappa_{B}(Q’’)=\{\begin{array}{ll}doi=\sum_{Q\subseteqq M}\overline{\alpha^{O}}(Q’’, Q’)-\delta_{|Q’|,\mathfrak{n}-1} if Q’’\subseteqq M(=0 if Q’’=P)\end{array}$$(= \alpha^{O}(Q’’,P)-\overline{\alpha^{O}}(Q’’,P)-\sum_{=}\delta(Q’’,Q))Q^{\mathfrak{n}-1\subset}M$$\kappa_{H}(Q’)=\{\begin{array}{ll}=1-\sum_{Q^{*-1}\subseteq M}\alpha(Q’,Q) if Q’\subseteqq M(=0 if Q’=P)\end{array}$$(= \zeta(Q’,P)-\delta(Q’,P)-\sum_{o^{n-1}\subseteq M}\alpha(Q’, Q))$,

where $\delta(Q’,Q’)=1;\delta(Q’’, Q’)=0$ if $Q”\subsetneqq Q’;\zeta(Q’’,Q’)=1$ if $Q”\subseteqq Q’;\partial(Q’’,Q’)=(-1)^{|Q’|-|Q’’|}\beta(Q’’,Q’)$, and ; and $\beta(Q’’,Q’)=0$ if $Q”\Subset Q’$ for each .
for$\beta=\alpha,$$\alpha^{O},$$\zeta$$\delta$$\beta$Corollary.
Mvm the generalization of the main result$(i.e.,$$\kappa_{B}=\overline{\alpha^{o}}0\kappa_{H}$$\kappa_{B}(Q’’)=\sum_{Q’}\overline{\alpha^{O}}(Q’’,Q’)\kappa_{H}(Q’))$

and$\alpha$$(i.e.,$$\overline{\alpha^{o}}=\delta$。

$\sum_{Q’}\alpha(Q’’,Q’)\overline{\alpha}^{T}(Q’,Q)=\delta(Q’’, Q))$,

we have$(i.e.,$$\alpha 0\kappa_{B}=\kappa_{H}$$\sum_{Q}\alpha(Q’,Q)\kappa_{B}(Q)=\kappa_{H}(Q’))$.

The following content was not described in the prosentation.
The theorem of Homma’s curvature.
Let$\tau(Q’,Q)d=\sum_{Q’\underline{\subseteq}Q’}(-1)^{|Q’|-|Q’’|}\alpha(Q’’,Q)of$$= \sum_{Q’’}\zeta(Q’’,Q’)\alpha(Q’’,Q)$$= \sum_{Q’’}\overline{\zeta}^{T}(Q’,Q’’)\alpha(Q’’, Q)$Then, for a$\dot{\alpha}mplexQ^{m}$and a face

$Q_{\neq}^{\prime\subset}Q$.

,$\tau(Q’,Q)=\frac{\mu_{m-1}(S(Q’Q’’))arrow}{\omega_{m-1}}$,

where $S(Q’Q”)arrow=$ { $\not\in’--fi\in S^{m-1}$ : $q\in Q’$ , Cl’ $\in Q’’$ }, $\omega_{m-1}=\mu_{m-1}(S^{m-1})$ , and the facesuch that the join of $Q’$ and $Q”$ is , and$Q$$\tau(Q,Q)=\{\begin{array}{ll}1 (ifm=0),0 (if m\geqq 1).\end{array}$$Q”\subsetneqq Q\dot{u}$91Remark.
The function
does not satisfy $\overline{(}^{T}(Q’’, Q’)=0$ if, so we cannot use the theorem above forspherical simplices and convex cones (we can only use it for Euclidean simplices).
$\overline{\zeta}^{T}$$Q”\subsetneqq Q’’$$\sum_{l=0}^{2}\frac{\theta^{1}}{2\pi}-\sum_{i}\frac{1}{2}+1=0$

for Homma’s curvature.
For a polyhdm, can we haveand a face$Co\iota\dot{\eta}ecture$$Q^{m}$$Q_{\neq}^{\prime\subset}Q$$= \emptyset\sum^{Q’\cap Q’’}(-1)^{|O’|+|Q’’|+1-m}\mu_{m-1}(S(QQ’’))arrow$$\tau(Q’, Q)=\frac{Q’’gQ}{w_{m-1}}?$’

and$\tau(Q, Q)=?\{\begin{array}{ll}1 (ifm=0),0 (if m\geqq 1).\end{array}$

REFERENCES.

[B] T. Banchoff, Oriticd Points and ouruature for embedd polyhdm, J. Differential Geom. 1 (1967),[H] T. Homma,, On $PL$ total curvatures, $Manlfold\epsilon-?bky_{Q}$ 1973 (Proc. Internat. Conf., Tbkyo, 1973) (1975), 111-116.
[加] 加藤十吉, 組合せ位相幾何学, 岩波書店, 岩波講座基礎数学幾何学 iv, 1976.
[M] P. McMullen, Non.
angle-sum $nlat|om$ for polyhdnl cones and PolytoPes, Math. Proc. Cambrldge Philoe. Soc.
78 (1975), 247-261.
[Sa] K. Sat6, Sphericalpolars, Quart. J. Math. 58 (2007), 107-126.
and$\epsilon d$.

$l2n\infty r$$\dot{\alpha}mpl:ce$$u_{\epsilon 1r}$

SAT\^o, Kenzi
DEPARTMENT OP MATHEMATICS, FACULTY OF ENCINBBRING, TAMAGAWA UNIVERSITY
6-1-1, $TAuAGAWA- GAKUBN$ , MACHIDA, TOKYO 194-8610, JAPANaddress:
tanagawa. ac. jp$E\sim ma|\downarrow$$k\cdot nzl0\cdot n_{G}$.

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