Alexandrov and differentiable sphere theorems

Assume the inherited Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ of the smooth comparison suppliers. Recorded orientation, not proved here. The following subjects are deferred; this remark supplies no proof prerequisite.

1. Alexandrov curvature. Lang, Definition 5.10, defines metric curvature bounds by local comparison of distances between corresponding side points in hinges (chord comparison), together with local existence of connecting segments. Theorem 5.12 identifies these bounds with sectional-curvature bounds on connected Riemannian manifolds. It does not assign sectional curvature to arbitrary length spaces. Lang's synthetic perimeter and diameter theorem (Theorem 5.17) assumes completeness, geodesicity, positive lower curvature, and an additional perpendicular-segment hypothesis at midpoints of segments longer than the model diameter. These synthetic results and their local metric structure are not developed here.

2. Sphere theorems beyond maximal diameter. Eschenburg, Theorem 11.1, gives a sphere homeomorphism under compactness, simple connectedness, positive sectional curvature and strict global quarter pinching. His Remark 11.2 records sharpness, and Remark 11.3 points to the separate Grove–Shiohama diameter theorem. Such results are deferred; a homeomorphism conclusion must not be silently strengthened to a diffeomorphism. The sphere conclusion proved in Cheng maximal diameter rigidity instead uses maximal diameter under a Ricci lower bound and concludes isometry with the round sphere. It supplies no pinching theorem.

3. Stability. Almost-equality statements and finiteness of topological or differentiable types require additional arguments; none is asserted or used in this pair.

The smooth Toponogov hinge comparison and Toponogov triangle comparison remain statements about Riemannian manifolds. The plan defers Alexandrov metric theory, Grove–Shiohama, pinching sphere theorems and stability. This unproved remark records that boundary and must not be used as a logical supplier.