I'm reading a paper and it references geodesics on singular spaces and I can't find a good definition for this.
Could someone provide a good definition?
2026-03-26 18:59:45.1774551585
What is a singular space in relation to differential geometry?
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From what you wrote, all one can say that it is a metric space which is not a Riemannian manifold with its distance function. Anything else is context dependent. Some authors mean an Alexandrov space , some mean a Riemannian metric with conical singularities , some mean a piecewise Riemannian cell complex ....
Edit. In the context of the paper you linked to, a singular submanifold is a real-analytic subset $M$ of $R^n$ equipped with the restriction of a Riemannian metric (you can even assume the standard one) from $R^n$ to $M$. The restricted Riemannian metric is singular at the singular points of $M$. The authors provide references (Lytchak, Mazzeo, et al.) for the detailed definitions, in their paper they consider a very special class of singularities, the cuspidal ones.