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A covariant 2-tensor field is linear over the ring of smooth functions. If $X, Y$ are vector fields and $f, h$ are smooth functions, then $g(fX, hY) = f h g(X,Y)$.

Conditions: $g$ is a covariant 2-tensor field.; $X, Y$ are vector fields.; $f, h$ are smooth functions.

A covariant 2-tensor field is linear over the ring of smooth functions. If $X, Y$ are vector fields and $f, h$ are smooth functions, then $g(fX, hY) = f h g(X,Y)$.

Conditions: $g$ is a covariant 2-tensor field.; $X, Y$ are vector fields.; $f, h$ are smooth functions.

A covariant tensor field takes vectors as inputs and outputs a scalar number. In the context of a Riemannian metric, it operates on pairs of vectors from the tangent space at a given point.

Conditions: The tensor field is covariant.; Inputs are vectors from the tangent space.; Output is a scalar.

A covariant tensor field takes vectors as inputs and outputs a scalar number. In the context of a Riemannian metric, it operates on pairs of vectors from the tangent space at a given point.

Conditions: The tensor field is covariant.; Inputs are vectors from the tangent space.; Output is a scalar.