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

from the fiber product of the tangent bundle of with itself to such that the restriction of to each fiber is a nondegenerate bilinear mapping

The mapping () is required to be continuous, and often continuously differentiable, smooth, or real analytic, depending on the case of interest, and whether can support such a structure.

Examples
Intrinsic definitions of a metric
স্থানাঙ্ক রূপান্তরের অধীনে পরিধির চাপের ধ্রুবক
Raising and lowering indices