Does space time cone live in spacetime manifold or in its tangent space? Does space time cone live in spacetime manifold or in its tangent space?
Given that it is defined by g(v, v) >= 0 I think it operates on velocity vectors which means space time cone is a structure defined on tangent space at every point.
However wikpedia says "In special and general relativity, a light cone is the path that a flash of light, emanating from a single event (localized to a single point in space and a single moment in time) and traveling in all directions, would take through spacetime."
This would imply the cone is in spacetime (manifold) itself.
I still think cone is defined in tangent space. Is this correct?
 A: It mostly depends on exactly how you define "light cone," but in the most common sense of the term, it lies in the spacetime manifold itself. In special relativity, the standard definition of the light cone emanating from a spacetime point $x_0$ is
$$\{ x_0 + \Delta x | g(\Delta x, \Delta x) = 0\}.$$
In general relativity the concept is more complicated: it's the set of points in spacetime that can be connected to the point $x_0$ by a null geodesic. (It's not generally a "cone" in the sense that it's not necessarily "flat," at least not in the naive sense of "flat".) But in general relativity, where we are often concerned with local physics, we often consider "infinitesimal" light cones of the form
$$\{ x_0 + dx | g_{\mu \nu}\, dx^\mu\, dx^\nu = 0\},$$
where the four-vectors $dx$ are infinitesimal. We can meaningfully add the vectors if we are considering a small enough region that we can neglect the curvature and treat is as locally flat. In this case, the vectors $dx$ live in the tangent space of the manifold at $x_0$.
