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In 3d Euclidean geometry, the velocity of any point of a rigid body is given by the cross product between its angular velocity and the position vector which links the instantaneous rotation center to the desired point.

Now let's imagine a 2d curved manifold. The notion of angular velocity can still be defined (the rigid body rotates about an axis perpendicular to the manifold tangent plane). However, the notion of position vector between two points is not defined here since the manifold is curved (unless its inhabitants are aware of the embedding 3d space, which is not necessary true).

So, how can the inhabitants of a curved 2d space use the angular velocity of a rigid body to determine the velocity on any of its points in an INTRINSIC way (i.e. using only their 2D curvilinear coordinates)?

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There is no notion of a rigid body if the space is curved. The best you can do is define a velocity $v^\mu$ vector for each point in the body and then construct the skew-tensor field $\omega_{\mu\nu} =\nabla_\mu v_\nu-\nabla_\nu v_\mu$. Here $\nabla_\mu$ is the usual covariant derivative.

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