Each cluster is about 6 nanometers wide and consists of about 4000 carbon atoms linked in graphite-like sheets that are given negative curvature by the inclusion of heptagons among the regular hexagonal pattern.
32.
The Kodaira dimension divides all " n "-dimensional varieties into " n " + 2 classes, which ( very roughly ) go from positive curvature to negative curvature.
33.
Subtracting a diamond-shaped dart produces positive curvature at the outer points of the diamond, and negative curvature at the middle points that are brought together ( good for the bust or back ).
34.
The sum of the angles of a triangle on a surface of positive curvature will exceed ?, while the sum of the angles of a triangle on a surface of negative curvature will be less than ?.
35.
The attractive force of gravity created by matter is due to a negative curvature of spacetime, represented in the rubber sheet analogy by the negatively curved ( trumpet-bell-like ) dip in the sheet.
36.
The system considers the motion of a free ( frictionless ) particle on a surface of constant negative curvature, the simplest compact Riemann surface, which is the surface of genus two : a donut with two holes.
37.
Geometrically, there is a very rough correspondence between Kodaira dimension and curvature : negative Kodaira dimension corresponds to positive curvature, zero Kodaira dimension corresponds to flatness, and maximum Kodaira dimension ( general type ) corresponds to negative curvature.
38.
In a 1993 paper Bowditch proved that five standard characterisations of geometric finiteness for discrete groups of isometries of Hadamard manifold of pinched ( but not necessarily constant ) negative curvature and of arbitrary dimension " n " e " 2.
39.
Agol's proof relies on the use of manifolds of pinched negative curvature and on Canary's trick of " diskbusting " that allows to replace a compressible end with an incompressible end, for which the conjecture has already been proved.
40.
Alternatively, the condition of negative curvature at a maximum is also equivalent to stating that the following logarithmic derivatives of the geometric means " G X " and " G ( 1 " X ) " are positive, since:
