I got the point of considering gravitational force as curvature on space-time fabric for bigger objects like stars, planets, blackholes. But my doubt is over the objects like us, what keeps us on this earth, the space-time curvature should not be able to hold the things on earth and on other planets as well. Then how does gravitational force work on the planet's surface?
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It works the same as anywhere else.
What we experience as "gravitational force" is in fact a consequence of the fact that geodesics ("straight lines" in the sense of paths through spacetime) are curved because spacetime is curved, and in the weak-field approximation this takes a form nearly-identical to Newton's law of gravitation.
Newton's shell theorem states that a spherically-symmetric shell of mass is equivalent to a point particle with the same mass at the shell's center, so we can imagine the Earth as a point mass with $M\approx5.97219\cdot10^{24}\,\mathrm{kg}$ and a massless barrier centered on that point with radius $R\approx6378\,\mathrm{km}$. The barrier here represents the Earth's surface; at that surface, the reaction force from touching the ground is exactly enough to keep particles from falling into the Earth, but surface of the ground, there is no reaction force and things fall towards the surface.
The thing about spacetime curvature is that for a distant observer it is impossible to distinguish between a sufficiently-weak Einsteinian gravitational field (i.e. one where the "force", which is in reality fictional, is just the curvature of straight paths because spacetime's all bent up) or a Newtonian one, where gravity is an actual force that holds everything together. So long as the Earth isn't a black hole or neutron star or wormhole (which I'm pretty sure it's not) where gravity becomes more extreme, the net result of geodesic curvature due to Einsteinian gravity and a Newtonian gravity force are approximately the same.
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