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If you look at the intrinsic geometry of a cone, there's a defect on the point of the cone known as a cone point. The only higher dimensional analogue I've heard of is what you get if you take the Cartesian product of a cone and the real line, where there's a similar defect along an entire line. What's the name of the defect at the point of a hypercone?

Also, there is an extension of the idea of a manifold called a cone point manifold, which is basically a manifold with cone points. The method used to define this doesn't work for the hypercone extension I'm looking for. What's it called, how is it defined, and what source can I cite?

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"conifold" is a good keyword to google for. –  Mariano Suárez-Alvarez Jul 20 '13 at 18:30

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