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:
“Wherever there are walls I shall inscribe this eternal accusation against Christianity upon them—I can write in letters which make even the blind see ... I call Christianity the one great curse, the one great intrinsic depravity, the one great instinct for revenge for which no expedient is sufficiently poisonous, secret, subterranean, petty—I call it the one immortal blemish of mankind....”
—Friedrich Nietzsche (1844–1900)
“The loosening, for some people, of rigid role definitions for men and women has shown that dads can be great at calming babies—if they take the time and make the effort to learn how. It’s that time and effort that not only teaches the dad how to calm the babies, but also turns him into a parent, just as the time and effort the mother puts into the babies turns her into a parent.”
—Pamela Patrick Novotny (20th century)