I am missing something in the space curvature thing

For an observer observing the crest of a light wave at a position and time , the crest of the light wave was emitted at a time in the past and a distant position . Integrating over the path in both space and time that the light wave travels yields:

In general, the wavelength of light is not the same for the two positions and times considered due to the changing properties of the metric. When the wave was emitted, it had a wavelength . The next crest of the light wave was emitted at a time

This is wiki explaining general relativity with regard to distant galaxies. There is something missing, I keep thinking.

The statement should read:

For an observers observing the crest of a light wave ar position r=0 and angle of incidence equal 0 at t = t now, the light wave was emmitted at a time t=then in the past at a distant position r = R and angle equal Angle. Where is the angle of incidence in a universe expanding at all directions? Then it should read, integrating over the path in both space, angle, and time.

The geodesic curvature explains:

This is also the idea of general relativity where particles move on geodesics and the bending is caused by the gravity.

OK, particles do not bend when space does. OK, if I were standing and counting particles arriving at my aperture, then sure, but I am not. There should be a double integral in that equation. The R^2 comes from the expansion of R along the point curve toward the source, as near as I can tell.

They are saying that over the path, energy density was stretched, so it takes more time, from our point of view, for the energy to arrive, so the energy we collect arrives at a lower rate, we measure lower energy density. Correct, but both along the line and along the area of our projected aperture. So, if distance expanded by a third, then our aperture shrunk by 80%.

Something seems odd, but it may just be me.