That's a lot harder to imagine because no 2D surface of constant negative curvature can be embedded in a 3D space, but mathematically it represents an infinite universe .-- talk ) 05 : 49, 17 May 2013 ( UTC)
42.
In practice, most of the interesting cases are surfaces with negative curvature, and are thus realized by a discrete lattice \ Gamma in the group \ operatorname { PSL } ( 2, \ mathbb { R } ) acting on the upper half-plane.
43.
Many map considered as chaotic are strongly mixing for some well-chosen invariant measure, including : the dyadic map, Arnold's cat map, horseshoe maps, Kolmogorov automorphisms, the geodesic flow on the unit tangent bundle of compact surfaces of negative curvature . ..
44.
Minkowski space is not endowed with a Euclidean geometry, and not with any of the generalized Riemannian geometries with intrinsic curvature, those exposed by the " model spaces " in hyperbolic geometry ( negative curvature ) and the geometry modeled by the sphere ( positive curvature ).
45.
The algebra \ mathfrak { sl } _ 2 ( \ mathbb { R } ) plays an important role in the study of hyperbolic plane, the simplest Riemann surface of negative curvature; by contrast, SL ( 2, C ) describes the automorphisms of the hyperbolic 3-dimensional ball.
46.
I think there's something in the wormhole article about stabilizing a wormhole with negative mass, which I think might give that negative curvature, but I really don't understand this stuff enough to say, so just count that as a request : doesn't negative curvature require negative mass?
47.
I think there's something in the wormhole article about stabilizing a wormhole with negative mass, which I think might give that negative curvature, but I really don't understand this stuff enough to say, so just count that as a request : doesn't negative curvature require negative mass?
48.
If the universe is the same everywhere ( homogeneous ) and there are no preferred directions ( isotropic ), then there are not many options for the symmetry group : they either live on a flat plane, or on a sphere with everywhere constant positive curvature, or on a Lobachevski plane with constant negative curvature.
49.
He begins by analysis of " the tip of a four-dimensional velocity vector " and notes Minkowski's equations where " both hypersurfaces provide a basis for a well-known model of non-Euclidean space of constant negative curvature, popularized by Helmholtz . " In fact it is known as the hyperboloid model of hyperbolic geometry.
50.
Recently, Ruoff and his group reported on a new carbon, potentially having regions of negative curvature carbon ( NCC ) with a remarkably high specific surface area of 3100 m?g " 1, and atom-thick carbon sp 2-bonded walls that define pores varying in diameter from about 0.6 to 5 nm.
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