A point of a manifold or metric space at which the neighborhood is locally isometric (or homeomorphic with compatible metric structure) to a metric cone over some base space; geometrically it appears as a 'tip' where the usual Euclidean structure fails and angular or radial metrics exhibit cone-like behavior.

Principle

Locally the metric can be written in polar-type coordinates as dr^2 + r^2 g_B + higher-order terms (for small r) where g_B is a metric on the base; the singularity is encoded by the base geometry and by an angle deficit or excess that modifies geodesic and curvature properties.

Demonstration

Flat cone in two dimensions: identify the plane by an angular sector of angle α with its sides glued; the apex is a conical singularity characterized by an angle 2π−α. Riemannian example: a metric on a surface with prescribed cone angles at finitely many points used in geometric constructions and orbifold theory.

Misapplication

Calling any sharp spike or unbounded curvature point 'conical' when the neighborhood lacks cone-coordinate structure. A cusp or fractal spike may be sharp but not modeled by a metric cone; likewise, removable coordinate singularities are not true conical points.

Consequence

Conical singularities alter geodesic completeness, change spectral properties of Laplace-type operators, introduce holonomy or angle deficits in geometric flows, and require adapted analytical tools (weighted spaces, model operators) for PDE analysis near the tip.

Reversal

A smooth regular point with Euclidean coordinate chart or a cusp singularity whose local model is not a cone; also surfaces with edge singularities (line singularities) rather than isolated conical tips.

Boundary

Pertains to isolated metric singularities modeled on cones; excludes extended edge singularities, distributed curvature singularities without cone structure, and purely topological identifications that do not preserve metric cone form.

Semantic Tension

Conical singularity versus cusp or edge: conical points have radial cone metrics and a well-defined base; cusps have different asymptotics (e.g., exponential narrowing), and edges are one-dimensional singular loci rather than isolated tips.

Synthesis

A conical singularity is an isolated metric defect where neighborhoods are modeled by a cone over some base manifold, characterized by a radial metric form and an angular structure that modifies geodesics, curvature, and spectral behavior compared with smooth points.