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2

You have misunderstood the nature of a singularity. The singularity at the centre of a black hole is not a point in space. Instead it is a place where spacetime becomes infinity curved, and it isn't possible to describe what happens there. Well, it's not possible using General Relativity, but we hope some future theory of quantum gravity will explain what ...

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Concerning (1), I strongly suggest you B. O'Neill's textbook about semi-Riemannian geometry. It is written for mathematicians interested in GR without any particular background in physics. The last chapters focus on the causal structure of spacetime and singularity theorems. There is a large part completely devoted to analyze Schwarzschild's metric and ...

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Well, the singularity does not concern the differentiable structure: Even around the tip of a cone (including the tip) you can define a smooth differentiable structure (obviously this smooth structure cannot be induced by the natural one in $R^3$ when the cone is viewed as embedded in $R^3$). Here the singularity is metrical however! Consider a $2D$ smooth ...

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