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In projective geometry, a line can be viewed by its projective line, which has only one point at infinity, and if a point keeps moving to $+\infty$ it will return to $-\infty$. On the other hand many integrals are taken over the real line. I wonder if there is a connection between two concepts. Would it be an integral over a circle or something? Would this be affect if the integral is replaced with a sum over $\mathbb Z$? I understand the function under integration is a member of $\mathbb R^\mathbb R$, and the series under summation is a member of $\mathbb R^\mathbb Z$

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  • $\begingroup$ Yes from $[x:y ] \mapsto [y:x]$ you see the topology of $\mathbb{P}^1(\mathbb{R})$ is of a circle and you can define an integral accordingly, for example with $\int_{\mathbb{P}^1(\mathbb{R})} f(u)du = \int_{-\pi/2}^{\pi/2} f([\tan x:1])dx$ $\endgroup$ – reuns Nov 6 '17 at 8:41
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The issue is choosing a metric (alternatively, a measure) on the real projective line with respect to which the integration is to be performed. It is natural to identify the real projective line with a circle in which case the natural metric is simply the least length of an arc joining the two points, i.e., the subtending angle $\theta$ in radians.

The way to define a distance between two points $A,B$ in the projective plane is to take the representing lines $a,b$ through the origin in the plane and set the distance between them be the angle between vectors $\alpha,\beta$ spanning $a,b$ respectively. Thus one would set the distance between $A$ and $B$ to be $\arccos \alpha \cdot \beta$ where "$\cdot$" is the scalar product, if $\alpha$ and $\beta$ are unit vectors.

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  • $\begingroup$ Oh, so the model of projective line depends on how the metric is defined on it? $\endgroup$ – Ooker Nov 6 '17 at 13:05
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    $\begingroup$ In projective geometry there is no distance; only relations of collinearity, concurrence, etc. If you want to integrate you have to add an additional piece of information to the picture, namely the metric and/or measure. In some cases there are fairly natural ones, as in the case of the circle. $\endgroup$ – Mikhail Katz Nov 6 '17 at 13:13
  • $\begingroup$ In the case of the metric induced from the euclidean $l^2$ norm, what exactly the circle we integrate? Can you give me an example? $\endgroup$ – Ooker Nov 6 '17 at 16:24
  • $\begingroup$ The Fourier series works for periodic functions, which can be imagined as functions in circle. Is there a connection between the two? $\endgroup$ – Ooker Nov 8 '17 at 15:07
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    $\begingroup$ To answer your question "what exactly the circle", the simplest way of putting it is to say that it is the circle parametrized by the angle $\theta$ appearing as $\arccos \alpha\cdot\beta$ as above, as $\theta$ runs from $0$ to $\pi$. $\endgroup$ – Mikhail Katz Nov 8 '17 at 15:14

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