The local isometry constraint prevents the objective function from going to infinity.
2.
If it were, then it would be a local isometry and would preserve Gaussian curvature.
3.
Gently bending a slice must then roughly maintain this curvature ( assuming the bend is roughly a local isometry ).
4.
If i, j \, \ ! are two neighbors, then the local isometry constraint that needs to be satisfied is:
5.
Since the homomorphism is a local isometry, we immediately conclude that to produce a uniform distribution on SO ( 3 ) we may use a uniform distribution on " S " 3.
6.
However, the Theorema Egregium of Carl Friedrich Gauss showed that for surfaces, the existence of a local isometry imposes strong compatibility conditions on their metrics : the Gaussian curvatures at the corresponding points must be the same.
7.
It can be shown that the G�del solution is, up to local isometry, the " only " perfect fluid solution of the Einstein field equation admitting a five-dimensional Lie algebra of Killing vectors.
8.
When " X " is equipped with this metric, " f " becomes a " local isometry ", that is the length of a curve equals to the length of its image.
9.
No mapping between a portion of a sphere and the plane can preserve both angles and areas . ( If one did, then it would be a local isometry and would preserve Gaussian curvature; but the sphere and disk have different curvatures, so this is impossible . ) This fact, that flat pictures cannot perfectly represent regions of spheres, is the fundamental problem of cartography.
10.
"Identical vertices " means that for any two vertices, there is a global isometry of the entire solid that takes one vertex to the other . pointed out a widespread error in the literature on Archimedan solids : some authors only require that the faces that meet at one vertex are related by a local isometry to the faces that meet at any other vertex, and incorrectly claim that there are 13 solids satisfying this definition.
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