Let $RP^n$ be the $n$-dim real projective space. I have the following four questions.

  1. What is the so called standard metric on $RP^n$?
  2. More generally, consider a metric space $M$ with an equivalent relation ~. Then there is a natural way to define a (pseudo)metric on the quotient space $M$\~. See Why are quotient metric spaces defined this way? for details. It seems to me that if we regard $RP^n$ as the quotient from $R^n$\{0}, then the induced pseudo metric is not a metric, however, if we regard $RP^n$ as the quotient from $S^n$, then the induced pseudo metric is a metric. Am I correct? Is there a contradiction?
  3. What is the standard measure on $RP^n$?
  4. How to in general define a measure on the quotient of a measure space? (similar to question 2)

Thank you for reading and thinking for my long questions. I put them in a logical order, however, I'm mostly interested in the third question. If you can't help me with all the questions, please try Q3 first.

  1. Apparently you ask about metric as two-point function, not about Riemannian metric in the projective space. The answer (consistent with the Riemannian metric) is: angle between lines through the origin in $\mathbb R^{n+1}$ (which represent points in $\mathbb RP^n$). The diameter of the space is $\pi/2$.

  2. Correct on both counts. Using the Euclidean metric on $\mathbb R^{n+1}\setminus \{0\}$ and collapsing along lines leads to a disaster: the quotient pseudo-metric is identically zero, because the Euclidean distance between any two lines through the origin is zero. This issue does not arise with $S^n/\{\pm I\}$, whether we use the restriction (chord) metric on $S^n$ or the intrinsic (arc) metric. The latter choice is more natural, and yields the metric described in 1.

  3. Pushing things from $S^n$ to $S^n/\{\pm I\}$ by the quotient map is a convenient and natural way to equip $\mathbb RP^n$ with various structures, including the measure. The pushforward of normalized Lebesgue measure on $S^n$ gives the standard probability measure on $\mathbb RP^n$.

  4. First, try pushing forward by the quotient map. Sometimes this does not work: for example, pushing the Lebesgue measure by the quotient map $\mathbb R\mapsto \mathbb R/\mathbb Z$ we get something ugly (which gives infinite measure to every nontrivial arc of the circle). In such a case, consider restricting the measure to a fundamental domain first.

  • $\begingroup$ Thank you so much for your answers. I want make sure I understood what you mean in Q3. So a set $E$ in $RP^n$ is measurable if the inverse of $E$ under the projection map is measurable, and the measure of $E$ is defined as the measure of its inverse. Am I right? $\endgroup$ – Tony B Jun 17 '13 at 15:02
  • $\begingroup$ @user16418 Yes, this is how pushforward measure works. Except we usually want a more concrete description of measurable sets. Clearly, the preimage of a Borel set under this projection is also Borel. This gives a Borel measure on the projective space, which we may decide to complete in the usual way. $\endgroup$ – ˈjuː.zɚ79365 Jun 17 '13 at 15:09

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for?Browse other questions tagged or ask your own question.