Isometric immersions with rectifiable geodesics. Q Han, M Lewicka, L Mahadevan, Rocky Mountain J. of Math.., Rocky Mountain J. of Math. 2024/08/25. Volume 54, Issue 4.

https://projecteuclid.org/journals/rocky-mountain-journal-of-mathematics/volume-54/issue-4/ISOMETRIC-IMMERSIONS-WITH-RECTIFIABLE-GEODESICS/10.1216/rmj.2024.54.1023.full