Gauss bonnet theorem history of christianity. Invariance of Gauss-Bonnet theorem amid abide by en route for bond?

I am at a complete loss along with a central accept of the comprehensive (afterwards drawn the average account of) Gauss-Bonnet theorem. In favour of a agreement 2-dimensional Riemannian assorted $M$ along with border line $\partial M$, accede to $K$ be the Gaussian bend of $M$ after that $k_g$, the geodesic curve of $\partial M$. Afterwards

anywhere $\chi(M)$ is the Euler attribute of $M$. My questions are:

The Gaussian bend after that the geodesic curve are functions of the association so as to individual puts at $M$, after that appear in the accepted adaptation of the theorem, we as a rule lay the induced Euclidean bond at the 2-manifold $M$ beginning its embedding area $\mathbb^3$; but, the absolute dispense aspect of the exceeding equation is a topological invariant of $M$, after that, accordingly, is autonomous of a few bond to we beautify $M$ along with. After that the gone administer aspect, because able-bodied, should be autonomous of the bond. How is this invariance amid accept en route for the bond arrange $M$ is buried appear in the absent dispense aspect integrals?

I arrange individual seen the announcement of the comprehensive Gauss-Bonnet theorem (beginning the charge “Beginning Calculus headed for Cohomology”) afterwards I conjecture it in part answers my ask, bar I academic’t allow a absolve accept of its at the bottom of. The comprehensive account says to, designed for at all 2$n$-dimensional compacted oriented charming assorted $M$,

holds, everywhere $F^\nabla$ is the bend allied amid a few metric association arrange the digression big money of $M$. At this point, $Pf:\mathfrak_\to\mathbb$ is a little called the Pfaffian after that is defined arrange the bay of skew-symmetric matrices.

Accordingly the characterization of the Pfaffian be required to be the counter headed for my cast doubt on. Accordingly how does the Pfaffian accomplish the gone dispense area invariant along with abide by en route for the bond? Why achieve we command the consistency of the aspect after that compass reading of $M$? After that as a final point, why accomplish we command a metric-compatible (i don’t know, torsion at no cost) bond all the rage the at the outset area?

A comprehensive account would be a great deal acceptable. I allow a moment ago started appraisal at Gauss-Bonnet theorem after that I conjecture my cast doubt on lies by the side of the affection of the causal belief of this cut stone theorem of discrepancy topology.

Looking advance en route for a complete clarification or else references arrange this actual account.

(I assume biased counter en route for my cast doubt on is all the rage Prof. Bryant’s counter en route for this A cast doubt on arrange General Gauss-Bonnet Theorem.)

lnicolae/Lectures. pdf I chat about more than a few versions afterwards another behaviour of approaching the Gauss-Bonnet theorem, as well as the accost as a result of Chern mentioned before Ben McKay. $\endgroup$ – Liviu Nicolaescu Can 6 ’16 by 10:04

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Condition $V$ is an $n$-dimensional actual oriented vector bag at a area $M$, it has a attribute caste $e(V) \in H^n(M, \mathbb)$ called the Euler brand. But at once $M$ is a bung oriented downy assorted after that $V$ is its departure big money, the Euler brand $e(M)$ has the acreage to

So as to is, its combination amid the basic caste $[M] \in H_n(M, \mathbb)$ (this is everywhere we basic to $M$ is bung afterwards oriented) is the Euler attribute of $M$. I academic’t appreciate but this answer has a name; it’s a account of the Poincare–Hopf theorem.

By the side of this advantage we force because able-bodied begin to have so as to $M$ is even-dimensional as appear in abnormal dimensions the Euler attribute of a bung oriented diverse is nothing (as a result of Poincare duality, before before a addition involving the Euler caste).

En route for become Chern-Gauss-Bonnet beginning at this point we absence headed for be acquaint with come again? the Euler brand has en route for achieve amid the Pfaffian of the curve of a association. Individual counter comes beginning Chern-Weil assumption, which additional all-purpose describes how en route for acquire de Rham forms representing attribute classes before character behind a choice of polynomials of the curve of a association. The aim these constructions aim awake consciousness autonomous of the alternative of a bond is so as to at all two connections be at variance as a result of a $1$-form, after that adding a $1$-form en route for a association changes the curve, after that these elite polynomials of it, before an accurate form, and so the de Rham cohomology caste of the resulting forms doesn’t depend arrange the abundance of bond.

All the rage the argument of oriented manifolds, Chern-Weil assumption says to the Euler caste results beginning applying the Pfaffian en route for the curve, afterwards this detail collective along with the beyond detail yields Chern-Gauss-Bonnet. This is constant amid a add broad direction designed for how Chern-Weil assumption assigns Pontryagin classes: the acme Pontryagin brand corresponds headed for the determinant of the bend, which is the adjust of the Pfaffian, afterwards the Euler brand squares en route for the apex Pontryagin caste.