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  1. Could you please provide me with the precise definition of isotropic curves of a conformal structure on a manifold $M$?

  2. If there is such a definition, then can I say the following: if $c$ is an isotropic curve corresponding to a Riemannian/Lorentzian metric $g$ on $M$, then $c$ is also an isotropic curve for any Riemannian/Lorentzian metric of the form $e^u\times g$, where $u\in C^{\infty}(M,\mathbb{R})$ is a smooth function on $M$?

3.One or two examples please?

Here is the semi-detail of the context I've come across the term 'isotropic':

Consider the $3$-dimensional anti-de-Sitter space $AdS_3^{*}=\{\vec{x}\in R^4: x_1^2+x_2^2-x_3^2-x_4^2=-1\}$, with its Lorentzian metric of signature (2,1), and its universal Lorentzian cover $AdS_3=\mathbb{D}\times \mathbb{R}$ with the Lorentz metric, which is a warped product of the hyperbolic metric $h$ on $\mathbb{D}$ and the Euclidean metric $dt^2$on $\mathbb{R}$, and is given by: $g=h-(\frac{1+r^2}{1-r^2})^2 dt^2$. Now, note that, if you consider $m=\frac{g}{(\frac{1+r^2}{1-r^2})^2}$, then the Lorentz metric $m$, which is conformal to $g$, extends to $\partial{\mathbb{D}}\times \mathbb{R}$.

The paper talks about the isotropic curves of this conformal Lorentz structure on $\partial{\mathbb{D}}\times \mathbb{R}$, and states that 'isotropic curves' of this structure foliates $\partial{\mathbb{D}}\times \mathbb{R}$.

I am unable to follow what they mean by these 'isotropic curves' in this context. A detailed explanation of the last paragraph will be greatly appreciated!

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Do you have some more context? I have come across the term isotropic in a possibly related context in semi-Riemannian geometry where it sometimes used to mean null/lightlike vectors, which are indeed preserved by conformal transformations... of course this is irrelevant if you are indeed talking about proper Riemannian manifolds. Regarding (2), if the curve really is defined from the conformal structure then of course it is invariant under conformal transformations. – Anthony Carapetis Aug 26 '13 at 0:27
Hello, thanks for your comment. I have come across the term in similar contect. I have edited my question, providing the details. Could you please take a look if you could? – Mathmath Aug 26 '13 at 1:25
up vote 3 down vote accepted

In Lorentzian geometry, isotropic/null/lightlike vectors $v$ are those that satisfy $g(v,v)=0$, and isotropic/null/lightlike curves $\gamma$ are those whose tangent vectors are everywhere isotropic; i.e. $g(\dot \gamma,\dot \gamma)=0$.

In your example it looks like $(\mathbb{D},h)$ is the Poincaré disc model $$\left(\{(x,y) \in \mathbb{R}^2 : x^2 + y^2 < 1\}, \frac{4}{(1-r^2)^2}(dx^2 + dy^2)\right)$$ where $r=\sqrt{x^2 + y^2}$ is the radial coordinate. Note that this does not extend smoothly to $\partial \mathbb{D}$ because $1-r^2=0$ there. However, if we modify it with the factor you mentioned, we get a $(1+r^2)^2$ instead, which can be extended to all of $\mathbb{R}^2$.

Likewise the product metric $$ m = \frac{4}{(1+r^2)^2}(dx^2 + dy^2) - dt^2$$ extends to all of $\mathbb{R}^2 \times \mathbb{R}$, and in particular to the surface $\partial \mathbb D \times \mathbb{R}$ (by taking the usual induced metric on a submanifold). Using polar coordinates $(r,\theta)$ on $\mathbb R ^2 $ so that the submanifold is $\{r=1\}$, this metric turns out to be simply $d\theta^2 - dt^2$ on the cylinder $S^1 \times \mathbb{R}$, which has two foliations by null curves: in coordinates $(\theta, t)$,

$$ \{\gamma_{\theta_0}^\pm(t) = (\theta_0 \pm t, t) : \theta_0 \in S^1\}.$$

Intuitively these are just helices on the infinite cylinder with angle $\pi/4$ to the horizontal/vertical axes.

Since the condition $g(\dot\gamma,\dot\gamma)=0$ is invariant under conformal changes (i.e. multiplying $g$ by a positive factor), the null curves are the same for any conformally related metric; so they are determined from the conformal structure only.

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Very informative and precise! Thank you a lot! – Mathmath Aug 26 '13 at 5:10

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