Intrinsic Invariants of Cross Caps

TL;DR

This paper investigates the intrinsic geometric invariants of cross caps, showing that three such invariants are fundamental and establishing the existence of extrinsic invariants, thereby deepening understanding of their geometric properties.

Contribution

The paper identifies three fundamental intrinsic invariants of cross caps and demonstrates the existence of extrinsic invariants, advancing the geometric theory of cross caps.

Findings

Existence of three fundamental intrinsic invariants for cross caps

Standard cross cap admits infinite-dimensional isometric deformations

Both intrinsic and extrinsic invariants of cross caps are established

Abstract

It is classically known that generic smooth maps of R^2 into R^3 admit only cross cap singularities. This suggests that the class of cross caps might be an important object in differential geometry. We show that the standard cross cap (u,uv,v^2) has non-trivial isometric deformations with infinite dimensional freedom. Since there are several geometric invariants for cross caps, the existence of isometric deformations suggests that one can ask which invariants of cross caps are intrinsic. In this paper, we show that there are three fundamental intrinsic invariants for cross caps. The existence of extrinsic invariants is also shown.