Let $P$ denote the quotient space obtained by the action of $\mathbb{Z}\backslash2\mathbb{Z}$ by the map $z\mapsto\frac{1}{z}$ on the riemann sphere $\hat{\mathbb{C}}$ (identified here with $\mathbb{C}\cup\left\{\infty\right\}$). I identify $P$ with the set:  $Q=\left\{ z\in\mathbb{C}:\left|z\right|&lt;1\right\} \cup\left\{ e^{it}:0\leq t\leq\pi\right\} $   that is to say, I use elements of $Q$ as the representatives for the equivalence classes in $P$. Note that this is a valid identification, seeing as every element of the orbit space of $z\mapsto\frac{1}{z}$ on $\hat{\mathbb{C}}$ is attested to by exactly one point in $Q$. I need a formula for a function $f:Q\times Q\rightarrow[0,\infty)$ such that $f$ is a metric on $P$. Just to be clear, I do not want an explanation of how to get such a formula; I want the formula.   An analogy for you if I have yet to make myself clear: suppose I was asking for the area of a square with side length $s$. The answers I have received so far for my question are akin to saying "multiply $s$ by itself" or "use the area formula for a square". The answer I am looking for is akin to saying "$s^{2}$". I need the formula. And please, no expressions with differentials, nor matrices, or any of that. I need to know how to compute the metric by using the complex numbers in the indicated set that I have identified with $P$.  As an aside, I found something called the "Fubini-Study metric" on wikipedia. I feel as if this might be close to what I need, but unfortunately, the use of notations and conventions of differential geometry and multilinear algebra makes the wikipedia article virtually unintelligible to me. If someone could give me a formula for how to compute this metric using the complex numbers in the set $Q$, it would be much appreciated—assuming that this Fubini-Study metric is even close to what I am actually asking for.   This is not for homework or anything. I am doing research (in analytic number theory), and I need to know a formula for such a metric so that I can go ahead and show whether or not a certain map is a contraction. I have been making stellar progress with my research, and so it is extremely frustrating to have my work ground to a halt for want of a simple formula.   For your answers, assume that I know nothing of differential geometry, riemannian geometry, tensors, riemannian metrics, pullbacks, metric tensors, projective geometry, multilinear algebra, differential forms, tangent spaces, tangent bundles, covering maps, stereographic projections (and so on). This is (obviously) not true, but I say so because I want to be absolutely certain that a prospective responder to my question understands the kind of answer I am looking for. Dumb it down.  If there is no such formula, then try to explain what I need to do as methodically and algorithmically as possible, and please, do not leave out any steps.   Please... help.
Let $P$ denote the quotient space obtained by the action of $\mathbb{Z}\backslash2\mathbb{Z}$ by the antipodal map $z\mapsto\frac{1}{z}$ on the riemann sphere $\hat{\mathbb{C}}$ (identified here with $\mathbb{C}\cup\left\{\infty\right\}$). I identify $P$ with the set:  $\left\{ z\in\mathbb{C}:\left|z\right|&lt;1\right\} \cup\left\{ e^{it}:0\leq t\leq\pi\right\} $   that is to say, I use elements of the above set as the representatives for the equivalence classes in $P$. I am looking for formula for a function $f:P\times P\rightarrow[0,\infty)$ such that $f$ is a metric on $P$. Specifically, I would like a formula for $f$ that I can evaluate by plugging in representative elements in the above set (or something like that, more or less).  Thanks in advance.  Edit: forgive me for sounding desperate, but I cannot make due with an explanation of how to obtain such a formula. I want the formula.   An analogy for you if I have yet to make myself clear: suppose I was asking for the area of a square with side length $s$. The answers I have received so far for my question are akin to saying "multiply $s$ by itself" or "use the area formula for a square". The answer I am looking for is akin to saying "$s^{2}$". I need the formula. And please, no expressions with differentials, nor matrices, or any of that. I need to know how to compute the metric by using the complex numbers in the indicated set that I have identified with $P$.
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