Is there a deeper connection between the following identies involving sums or integrals over a closed path (resp. circle) resp. the area enclosed – e.g. a general principle that is underlying all of them?

- Euler's totient is the Fourier transform of the greatest common divisor function:

$$\sum_{k=1}^n \operatorname{gcd}(k,n)\cdot e^{i2\pi k/n} = \varphi(n)$$

- The sum of the $n$-th roots of unity is $0$:

$$\sum_{k=1}^n e^{i2\pi k/n} = 0$$

$$\oint_\gamma f(z)dz = 0 $$

$$\oint_{\gamma} f(z)dz = 2\pi i \sum_{k=1}^n\operatorname{I}(\gamma,a_k)\operatorname{Res}(f,a_k)$$

- The Gauss-Bonnet theorem:

$$\int_{\partial M}k_g\ ds = \int_M K\ dA + 2\pi\chi(M)$$

For $K\equiv 0$:

$$\int_{\partial M}k_g\ ds = 2\pi\chi(M)$$

$$\int _{M}{\mbox{Pf}}(\Omega )=(2\pi )^{n}\chi (M)$$

Or is there possibly no deep connection between (all of) them, and the analogies are superficial?

Furthermore: Is this selection too arbitrary, and examples of such identities abound, actually?

A connection between 2, 3, 4 is demonstrated here.