I De Sitter Universe: Divergent Parallel Lines?

[dagmar]

The hyperboloid with equation: $z^2=x^2+y^2-1$, embedded in standard 3-D Minkowski space ( +, +, - ) so that $z^2$ is negative, has positive Gaussian curvature equal to 1 ( as found directly from its metric: $ds^2 = \sqrt{ -dτ^2+(Coshτ)^2 dθ^2 }$ induced from the ambient Minkowski metric ) and is representing a 2-dimensional De Sitter Universe submanifold.

Nevertheless, there exist parallel geodesic lines ( the hyperbolas on the surface perpendicular to the circle $x^2+y^2=1$ ) which diverge instead of converging as intuition has it, for surfaces of constant positive curvature.

Is this a characteristic of the signature of the embedding space to have such divergent parallels, contrary to the all convergent parallels of a constant positive Gaussian curvature surface embedded in a ( +, +, .., + ) space, say?

 

[PeterDonis]

there exist parallel geodesic lines ( the hyperbolas on the surface perpendicular to the circle $x^2+y^2=1$ ) which diverge instead of converging as intuition has it, for surfaces of constant positive curvature.

These hyperbolas are timelike, correct? The intuition you refer to is only valid in a manifold with a positive definite metric, or for spacelike geodesics in a manifold with Lorentzian metric.

 

[dagmar]

These hyperbolas are timelike, correct? The intuition you refer to is only valid in a manifold with a positive definite metric, or for spacelike geodesics in a manifold with Lorentzian metric.

So this settles my question. Yes, they are time-like.
I take your word Peter, that if the Gaussian curvature is positive and everywhere constant in a manifold with a positive definite metric then parallel geodesic lines always converge.
In my case the metric is negative, so this is not the case. And indeed, like you said spacelike geodesics converge in my manifold example. They are just the ellipse-like lines generated when the hyperboloid is cut by planes passing through the origin (0,0,0) at an angle less than 45 degrees.
Thank you.

 

[PeterDonis]

if the Gaussian curvature is positive and everywhere constant in a manifold with a positive definite metric then parallel geodesic lines always converge.

Yes, AFAIK that's correct.