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
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