Euler Characteristic
المؤلف:
Alexandroff, P. S.
المصدر:
Combinatorial Topology. New York: Dover, 1998.
الجزء والصفحة:
...
31-5-2021
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Euler Characteristic
Let a closed surface have genus 
. Then the polyhedral formula generalizes to the Poincaré formula
 |
(1)
|
where
 |
(2)
|
is the Euler characteristic, sometimes also known as the Euler-Poincaré characteristic. The polyhedral formula corresponds to the special case 
.
The only compact closed surfaces with Euler characteristic 0 are the Klein bottle and torus (Dodson and Parker 1997, p. 125). The following table gives the Euler characteristics for some common surfaces (Henle 1994, pp. 167 and 295; Alexandroff 1998, p. 99).
| surface |
 |
| cylinder |
0 |
| double torus |
 |
| Klein bottle |
0 |
| Möbius strip |
0 |
| projective plane |
1 |
| sphere |
2 |
| torus |
0 |
In terms of the integral curvature of the surface 
,
 |
(3)
|
The Euler characteristic is sometimes also called the Euler number. It can also be expressed as
 |
(4)
|
where 
is the 
th Betti number of the space.
REFERENCES:
Alexandroff, P. S. Combinatorial Topology. New York: Dover, 1998.
Armstrong, M. A. "Euler Characteristics." §7.3 in Basic Topology, rev. ed. New York: Springer-Verlag, pp. 158-161, 1997 Coxeter, H. S. M. "Poincaré's Proof of Euler's Formula." Ch. 9 in Regular Polytopes, 3rd ed. New York: Dover, pp. 165-172, 1973.
Dodson, C. T. J. and Parker, P. E. A User's Guide to Algebraic Topology. Dordrecht, Netherlands: Kluwer, 1997.
Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 635, 1997.
Henle, M. A Combinatorial Introduction to Topology. New York: Dover, p. 167, 1994.
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