Bends in Space-Time vs Invariance

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Imagine a beam of light being turned on at the surface of a massive body such as a neutron star, the beam of light travels along a geodesic path towards a mirror located at radius r from the planet's surface, when the light beam hits the mirror, the light bounces back to the observer located at the source of the original beam.

My question is does the speed of light (c), when traveling along a geodesic from a massive body to an observer vary according to bends in space-time? (as if light has traveled a greater distance according to the curvature of space-time) or does light travel at the speed of light (c) in a geodesic no matter what the curvature of space-time is?

In the example, would the observer's observations for the time the light beam to reach his eyes be the same as if it were located at the distance (r) in flat space-time?
 
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