# Saddle Shaped Universe

The universe, as described by FLRW metric, if $$k = -1$$ is clearly a 2 sheet 3-hyperboloid described by $$x^2+y^2+z^2-w^2=-R^2$$. So where does the more common saddle shaped picture of the open universe come from?

• I think do not understand your question. The saddle in the picture is a 2D representation of the spatial section of the FLRW universe for $k=-1$. Where is the problem? Is the fact that we see only one one sheet in the picture? Indeed, there is the further assumption that the spatial sections are connected, so only one sheet is considered...That is not the only assumtion: isometric identifications accorndig to a discrete subgroup of isometries are in principle also possible... Commented Jul 3 at 9:45
• The 2D representation (dropping one of the dimensions) should still be a hyperboloid described by $x^2 + y^2 - w^2 = -R^2$. Referring to the FLRW metric, keeping only r and theta it is $ds^2 = \frac{1}{1+r^2}dr^2 + r^2 d\theta ^2$. Commented Jul 3 at 10:47
• It is just a change of coordinates... Commented Jul 3 at 11:33
• A saddle is a different shape. It’s not a hyperboloid at all. Commented Jul 3 at 12:39

## 1 Answer

It is simply an analogy (and not a very precise one) to illustrate that even in our familiar 3D space there are surfaces with negative curvature at all points. What is drawn in the illustration may be part of a one-sheet hyperboloid $$x^2+y^2-z^2=1$$ or may perhaps be a hyperbolic paraboloid $$z = x^2 - y^2$$.

• @ValterMoretti Indeed - which is why I said the saddle surface image is a loose analogy, not a precise one. Commented Jul 3 at 16:37