I am having trouble understanding the nature of the metric tensor field on spacetime manifolds.
In particular, a Riemannian manifold $(M,g)$ is defined as a real smooth manifold $M$ equipped with an inner product $g_p$ on the tangent space $T_pM$ at each point $p$ that varies smoothly from point to point in the sense that if $X$ and $Y$ are vector fields on $M$, then $p \mapsto g_p(X(p),Y(p))$ is a smooth function. The family $g_p$ of inner products is called a Riemannian metric tensor.
But in my physics classes, I often hear the equation
$ds^2 = -dt^2 + dx^2 + dy^2 + dz^2$
referred to as a "metric."
Is it a Riemannian metric?
Can $ds^2 = -dt^2 + dx^2 + dy^2 + dz^2$ be written as a tensor field?