  We consider a spherically symmetric line element which admits either a black
hole geometry or a wormhole geometry and show that in both cases the apparent
horizon or the wormhole throat is partially characterized by the zero-set of a
single curvature invariant. The detection of the apparent horizon by this
invariant is consistent with the geometric horizon detection conjectures and
implies that it is a geometric horizon of the black hole, while the detection
of the wormhole throat presents a conceptual problem for the conjectures. To
distinguish between these surfaces, we determine a set of curvature invariants
that fully characterize the apparent horizon and wormhole throat. Motivated by
this result, we introduce the concept of a geometric surface as a
generalization of a geometric horizon and extend the geometric horizon
detection conjectures to geometric surfaces. As an application, we employ
curvature invariants to characterize three important surfaces of the line
element introduced by Simpson, Martin-Moruno and Visser which describes
transitions between regular Vaidya black holes, traversable wormholes, and
black bounces.
