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The definition is the same as in Mathematics and, I suppose, elsewhere:

it is a topological space such that the vicinity of each point is homeomorphic to a disk in $\mathbb{R}^n$ (note, that dimension has to be the same for all points $x$). This requirement is important, since not every imaginable geometric object satisfies this requirement:

  • Sphere is a manifold, since one can draw a tangent plane in the vicinity of any point. It is "locally flat" and even we humans see Earth flat, since its radius of curvature is much greater than the visible distance (let the proponents of the flat Earth theory forgive me).

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  • Two intersecting lines are not a manifold, since for any point, except for the intersection we have a 1-dimensional space and for the intersection point, there are two non-collinear vectors belonging to them.

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As a good material on this topic, I recommend this lecture from the recent workshop.