Titre: Displacement of an Extended Body by Metric Deformation: The Geometric Displacement Theorem Abstract: Let (M, g) be a compact Riemannian manifold without boundary and M an extended body with smooth mass density > 0, initially at rest. When the metric evolves along a smooth family g (t) with g (0) = g, the Karcher center of mass p (t) of is displaced — without any applied force and without initial velocity. We prove the Geometric Displacement Theorem: d₆ (p (T), p (0) ) C (, , g) D_ where D: = ₀T (|ġ|g (t) + |g (t) ġ|g (t) ) dVg (t) dt is the deformation functional restricted to. A matching lower bound holds under a non-degeneracy condition, giving the equivalence c D dg (p (T), p (0) ) C D. In particular, p (t) p (0) if and only if g is static on — a Riemannian generalization of Newton's first law. Applications to the Ricci flow and to cosmological spacetimes via the ADM decomposition are given, including an exact closed-form expression for D in FLRW geometry.
Judicael Brindel (Sun,) studied this question.