Gauss Formula - Bonn

The Gauss – Bonnet formula relates the Euler characteristic of a surface with its Gaussian curvature and the geodesic curvature of its boundary.

Content

Wording

Let be $\Omega$ ${\ displaystyle \ Omega}$ Is a compact two-dimensional oriented Riemannian manifold with a smooth boundary $\partial \Omega$ ${\ displaystyle \ partial \ Omega}$ . Denote by $K$ ${\ displaystyle K}$ gaussian curvature $\Omega$ ${\ displaystyle \ Omega}$ and through $k_{g}$ ${\ displaystyle k_ {g}}$ geodesic curvature $\partial \Omega$ ${\ displaystyle \ partial \ Omega}$ . Then

\int \limits _{\Omega }K\;d\sigma +\int \limits _{\partial \Omega }k_{g}\;ds=2\pi \chi (\Omega ),

{\ displaystyle \ int \ limits _ {\ Omega} K \; d \ sigma + \ int \ limits _ {\ partial \ Omega} k_ {g} \; ds = 2 \ pi \ chi (\ Omega),}

Where $\chi (\Omega )$ ${\ displaystyle \ chi (\ Omega)}$ - Euler characteristic $\Omega$ ${\ displaystyle \ Omega}$ .

In particular, if $\Omega$ ${\ displaystyle \ Omega}$ no border, we get

\int \limits _{\Omega }K\;d\sigma =2\pi \chi (\Omega )

{\ displaystyle \ int \ limits _ {\ Omega} K \; d \ sigma = 2 \ pi \ chi (\ Omega)}

If the surface is deformed, then its Euler characteristic does not change, while the Gaussian curvature can change pointwise. However, according to the Gauss-Bonnet formula, the integral of the Gaussian curvature remains the same.

History

A special case of this formula for geodesic triangles was obtained by Friedrich Gauss ^[1] . ^[2] and Jacques Binet independently generalized the formula to the case of a disk of a bounded arbitrary curve; Binet did not publish an article on this subject, but Bonnet mentions this on page 129 of his "Mémoire sur la Théorie Générale des Surfaces". For non-simply connected regions, the formula appears in the work of Walter von Dieck ^[3] . The modern formulation is given by William Blaschke ^[4] .

Variations and generalizations

The Gauss – Bonnet formula naturally generalizes to domains with a piecewise-smooth boundary. If at the break point $P$ $i$ {\ displaystyle P_ {i}} tangent vector $τ$ {\ displaystyle {\ boldsymbol {\ tau}}} swivels $ϕ$ $i$ {\ displaystyle \ phi _ {i}} towards the area $Ω$ {\ displaystyle \ Omega} (can be a positive or negative number), then the formula generalizes to this:
$\int \limits _{\Omega }Kd\sigma +\int _{L}k_{g}ds+\sum _{i}\phi _{i}=2\pi \chi (\Omega )$ ${\ displaystyle \ int \ limits _ {\ Omega} Kd \ sigma + \ int _ {L} k_ {g} ds + \ sum _ {i} \ phi _ {i} = 2 \ pi \ chi (\ Omega)}$
- To display this formula, the area $\Omega$ ${\ displaystyle \ Omega}$ need to be approximated by a region that has smoothed corners. Then the radius of the curve at the corners is directed to zero.

The generalized Gauss - Bonnet formula is a generalization of the Gauss - Bonnet formula to higher dimensions.

The Kon-Vossen inequality is a generalization of the Gauss-Bonnet formula to non-compact surfaces.

According to the Gauss-Bonnet formula, any triangle on a full surface of non-negative Gaussian curvature has a sum of angles of at least $\pi$ ${\ displaystyle \ pi}$ . The Toponogov comparison theorem refines this inequality.

Links

↑ CFGauss, Disquisitiones generales circa superficies curvas, Commentationes Societatis Regiae Scientiarum Gottingesis Recentiores. Volume VI, pp. 99–146.
↑ Bonnet, 1848 'Mémoire sur la Théorie Générale des Surfaces', J. École Polytechnique 19 (1848) pp. 1-146
↑ von Dyck W. Beiträge zur analysis situs. Math Ann 32: 457-512 (1888)
↑ Wilhelm Blaschke, Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie, 1921

S.E. Stepanov, Gauss — Bonnet Theorem, Coolant, 2000, No. 9, p. 116-121.
Wu, Hung-Hsi. "Historical development of the Gauss-Bonnet theorem." Science in China Series A: Mathematics 51.4 (2008): 777-784.

[1] CFGauss, Disquisitiones generales circa superficies curvas, Commentationes Societatis Regiae Scientiarum Gottingesis Recentiores. Volume VI, pp. 99–146.

[2] Bonnet, 1848 'Mémoire sur la Théorie Générale des Surfaces', J. École Polytechnique 19 (1848) pp. 1-146

[3] von Dyck W. Beiträge zur analysis situs. Math Ann 32: 457-512 (1888)

[4] Wilhelm Blaschke, Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie, 1921