Normal Curvature and the Projective Systole

What the math gives to ML

The paper gives a concrete, scale-normalized way to measure extrinsic bending of an immersion and identifies the Veronese map as the sharp model for real projective space. Its most transferable construction is the antipodal-invariant quadratic embedding of a vector into the traceless symmetric-matrix space, which converts projective inputs into structured second-order features while preserving rotational symmetry. This can be used as a front end for data whose semantics identify x and -x, or as a benchmark and regularizer for learned embeddings. The curvature inequality also suggests penalizing excessive second fundamental form in neural manifolds, although that extension is less immediately robust than directly using the Veronese representation.