##  [Conical Singularity](/conical-singularity-0) 

 Definition

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

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

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

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

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

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

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

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

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.