- #1
dagmar
- 30
- 7
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?
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?