Intrinsic Definitions of A Metric
The notion of a metric can be defined intrinsically using the language of fiber bundles and vector bundles. In these terms, a metric tensor is a function
-
(5)
from the fiber product of the tangent bundle of M to R such that the restriction of g to each fiber is a nondegenerate bilinear mapping
The mapping (5) is required to be continuous, and often continuously differentiable, smooth, or real analytic, depending on the case of interest, and whether M can support such a structure.
Read more about this topic: Metric Tensor
Famous quotes containing the words intrinsic and/or definitions:
“To have that sense of one’s intrinsic worth which constitutes self-respect is potentially to have everything: the ability to discriminate, to love and to remain indifferent. To lack it is to be locked within oneself, paradoxically incapable of either love or indifference.”
—Joan Didion (b. 1934)
“Lord Byron is an exceedingly interesting person, and as such is it not to be regretted that he is a slave to the vilest and most vulgar prejudices, and as mad as the winds?
There have been many definitions of beauty in art. What is it? Beauty is what the untrained eyes consider abominable.”
—Edmond De Goncourt (1822–1896)