Quadratic Forms vs. Metric Tensors: A Foundational Comparison

This paper revisits the distinction between quadratic forms and metric tensors. In threedimensional Euclidean geometry, quadratic forms and metrics coincide only in Cartesian coordinates, while in other orthogonal coordinate systems Lamé coefficients must be introduced. By contrast, in Gauss surface geometry quadratic forms alone cannot represent intrinsic curvature; the induced metric is required. We argue that the metric tensor should be interpreted as an invariant property of space, encoding local homogeneity or inhomogeneity and determining curvature.