The metric tensor tells you how to measure distances and angles in a given space. In flat space (like Euclidean geometry), the metric is just the dot product. But in general relativity, it encodes the curvature of spacetime. Because it is a bilinear form, it ensures that the inner product behaves linearly when combining vectors, preserving the fundamental structure of geometry.

bilinear form(metric tensor) just combination of convectors. also heard projection operator, cross operator, outer operator and density operator, how are they related?

projection operator:

translation operator:

function of operators: commutators are involved in function of operators!