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I understand how there can be no center if the universe has positive curvature. But if the universe is saddle-shaped and non-infinite, surely there has to be a
by cout 5y ago
I understand how there can be no center if the universe has positive curvature. But if the universe is saddle-shaped and non-infinite, surely there has to be a center?
- effie 5y agoI think not; consider pseudosphere, there is no obvious center. Its embedding in 3D space has infinite dimensions but the object itself has finite surface area. https://en.wikipedia.org/wiki/Pseudosphere https://en.wikipedia.org/wiki/Pseudosphere If you mean object with finite dimensions and negative curvature and no center, there seems to be such a thing, see Fig. n066200e: https://encyclopediaofmath.org/wiki/Negative_curvature,_surface_of https://encyclopediaofmath.org/wiki/Negative_curvature,_surf... EDIT Rozendorn's idea seems to be still an open question: https://mathoverflow.net/questions/52851/the-geometry-of-nadirashvilis-complete-bounded-negative-curvature-surface https://mathoverflow.net/questions/52851/the-geometry-of-nad...
- feoren 5y agoRather than think "surface of a balloon inflating", think "surface of an (infinite?) rubber sheet being uniformly stretched". It does intuitively seem like if there is 0 curvature and no "center", then the universe must be infinite though ...