GR Metric Tensor Rank 2: Quadratic vs Shear Forces

[DiracPool]

Is the metric tensor a tensor of rank two simply because the line element (or equivalent Pythagorean relation between differential distances) is "quadratic" in nature? This would be in opposition to say, the stress tensor being a tensor of rank two because it has to deal with "shear" forces. I always thought GR dealt with tensors because we were dealing with multiple physical phenomena along the same space-time axes, such as stress-pressure, current, energy density, charge etc.

 

[Nugatory]

A tensor of rank N maps N vectors to a scalar and N-1 vectors to a vector.

The metric tensor is of rank two because it maps two vectors to a scalar.