My main research interests are in Differential Geometry. For the past two decades I have been interested in Lie sphere geometry applied to problems of Dupin hypersurfaces in spheres and isoparametric hypersurfaces, and to surfaces in space (that is, in three-dimensional space forms). In recent years I've been interested in the Bonnet problem.
The following is a list of my books and papers. There are some links to preprints (as pdf files).
1. Homogeneous Einstein Spaces of Dimension Four, J. Diff. Geom. 3 (1969), pp. 309-349 (Dissertation).
2. The scalar curvature of left-invariant Riemannian metrics, Indiana University Math. J. 29 (1971), pp. 1125 - 1144.
3. Einstein metrics on principal fire bundles, J. Diff. Geom. 8 (1973), pp. 599 - 614.
4. Imbeddings of Stiefel manifolds into Grassmannians, Duke Math. J. 42 (1975), pp. 397 - 407. Reprint
5. Higher order contact of submanifolds of homogeneous spaces, Lecture Notes in Math. 610, Springer-Verlag, New Nork, 1977, 166 pages. Copies available
6. Deformation of submanifolds of homogeneous spaces, J. Diff. Geom. 16 (1981), pp. 213-246.
7. Projective deformation and biholomorphic equivalence of real hypersurfaces, Ann. Global Anal. and Geom. 1 (1983), pp. 1 - 34.
8. Second order projective deformation of real hypersurfaces, Tensor, N.S. 39 (1983), pp. 135 - 140.
9. Minimal surfaces in spheres by the method of moving frames, Revue de l'Insitute E. Cartan 12 (1988), Nancy, France (with M. Rigoli). Reprint
10. Minimal surfaces in spheres, Rend. Sem. Mat. Univers. Politec. Torino, Special Volume (1983), pp. 75 - 98 (with M. Rigoli). Reprint
11. A class of harmonic maps from surfaces into real Grassmannians, Rend. Sem. Mat. Univers. Politec. Torino, Special Volume (1983), pp. 99 - 116 (with M. Rigoli). Reprint
12. Quantization of curvature of harmonic two-spheres in CPn, preprint (1986), 41 pages, preliminary version of [15] (with M. Rigoli).
13. Differential systems and isometric embeddings, Annals of Math. Study 114, Princeton Universtiy Press, 1987, 225 pages (with P. Griffiths). Copies available
14. Holomorphic curves in the complex quadric, Bull. Australian Math. Soc. 35 (1987), pp. 125 - 147 (with M. Rigoli and K. Yang).
15. On conformal minimal immersions of S2 into C Pn, Math. Ann. 279 (1988), pp. 599-620, with J. Bolton, M. Rigoli and L. Woodward.
16. Harmonically immersed surfaces of Rn, Trans. Amer. Math. Soc. 307 (1988), pp. 363 - 372 (with M. Rigoli). Correction, Trans. Amer. Math. Soc. 311 (1989), pp. 425 - 428.
17. Harmonic maps, twistor spaces and twistor lifts, Fibre Bundles: Their Use in Physics, eds. J.T. Ezin and A. Verjovski, World Scientific Press, Singapore, 1988, pp. 11 - 22 (with M. Rigoli).
18. Harmonic Gauss maps, Pacific J. Math. 136 (1989), pp. 261 - 282 (with M. Rigoli).
19. On the isotropy of compact minimal surfaces in CPn, Math Z. 200 (1989), pp. 169 - 180 (with M. Rigoli).
20. Twistor and Gauss lifts of surfaces in four manifolds, Recent Developments in Geometry 101, eds. S.-Y. Cheng, H. Choi and R. Greene, Amer. Math. Soc., Providence, RI, 1989, pp. 197 - 232 (with M. Rigoli). Reprint
21. Review of Einstein Manifolds by A. Besse, Springer - Verlag, 1987, Bull. Amer. Math. Soc. 20 (1989), pp. 142 - 145.
22. Neutral surfaces in neutral four-manifolds, Le Matematiche 45 (1990), pp. 407 - 443 (with M. Rigoli). Reprint
23. Einstein metrics on circle bundles, Proc. Symposia in Pure Math. 54 (1993), Part 2, pp. 331 - 336 (with M. Rigoli). Reprint
24. Rigidity of hypersurfaces in complex projective space, Ann. Sci. Ecole Norm. Sup. 27 (1994), pp. 227 - 248 (with E. Musso).
25. Families of flat minimal tori in CPn, J. Diff. Geom. 42 (1995), pp. 113-132 (with R. Liao).
26. Isoparametric functions and flat minimal tori in CP2, Proc. Amer. Math. Soc 123 (1995), pp. 2849-2854 (with Q-S. Chi and R. Liao).
27. Dupin hypersurfaces, Geometry and Topology of Submanifolds, VII, eds. F. Dillen et al., World Scientific, River Edge, New Jersey, 1995, pp. 100 - 105 (with T. Cecil).
28. Irreducible Dupin hypersurfaces with three principal curvatures, Inventiones Mathematicae 132 (1998) pp. 121-178 (with T. Cecil).
29. Jump-Diffusion Markov Processes on Orthogonal Groups for Object Pose Estimation, J. of Statistical Planning and Inference, 103, Issues:1-2 (2002) pp. 15-37 (with A. Srivastava, U. Grenander, & M.I. Miller).
30. Dupin hypersurfaces with four principal curvatures, Geometriae Dedicata 79 (2000) pp. 1-49 (with T. Cecil).
31. Using MATLAB in Calculus, Prentice Hall, 2000, 174 pages.
32. Arithmetic for Teachers. With Applications and Topics from Geometry, American Mathematical Society, Providence, RI, 2003, xiii + 383 pages.
33. 150 Years of Mathematics at Washington University in St. Louis, Amer. Math. Soc., Contemporary Mathematics 395, (2006) (edited with S.G. Krantz).
34. Isoparametric hypersurfaces with four principal curvatures, Annals of Math, 166 (2007), pp. 1-76 (with T. Cecil and Q-S. Chi). Reprint
35. Dupin hypersurfaces with four principal curvatures, II, Geometriae Dedicata, 128 (2007), pp. 55-95 (with T. Cecil and Q-S. Chi).
36. Classifications of Dupin Hypersurfaces, in F. Dillen and I. Van de Woestyne, eds., Pure and Applied Differential Geometry, PADGE 2007, Shaker Verlag, Aachen, (2007), 48-56 (with T. Cecil and Q.-S. Chi).
37. On Kuiper's question whether taut submanifolds are algebraic, Pac. J. of Math. 234 (2008), No. 2, pp. 229-247 (with Q.-S. Chi and T. Cecil). Reprint
38. The geometric Cauchy problem for the membrane shape equation, J. Phys. A 47 (2014), no. 49, 495201, 22 pp. (with E. Musso and L. Nicolodi). Preprint on arXiv
39. Remembering Shoshichi Kobayashi, Notices of the AMS, 61, No. 11, December 2014, pp 1322 - 1332, (coordinating editor).
40. Dupin hypersurfaces in Lie sphere geometry, pages 383-394 in Geometry and Analysis on Manifolds. In Memory of Professor Shoshichi Kobayashi. Editors: Ochiai, T., Mabuchi, T., Maeda, Y., Noguchi, J., Weinstein, A. Volume 308 of the series Progress in Mathematics, Birkhauser, 2015. Preprint on arXiv
41. Surfaces in Classical Geometries. A Treatment by Moving Frames. xiii + 571 pages. Universitext, Springer, 2016 (with E. Musso and L. Nicolodi). Link
42. Compact surfaces with no Bonnet mate, J. Geom. Anal. 28, pp 2644 - 2652 (2018) DOI 10.1007/s12220-017-9924-y (with E. Muuso and L. Nicolodi). Reprint
43. Lie sphere geometry in R3, pages 223 - 255 in Geometry of Lagrangian Grassmannians and Nonlinear PDEs, Banach Center Publications Volume 117, Warsaw 2019.
Last modified: Thursday 12 November 2020.