In differential geometry, the Carathéodory conjecture is a mathematical conjecture attributed to Constantin Carathéodory by Hans Ludwig Hamburger in a session of the Berlin Mathematical Society in 1924. Carathéodory did publish a paper on a related subject, but never committed the Conjecture into writing. In, John Edensor Littlewood mentions the Conjecture and Hamburger's contribution  as an example of a mathematical claim that is easy to state but difficult to prove. Dirk Struik describes in  the formal analogy of the Conjecture with the Four Vertex Theorem for plane curves. Modern references to the Conjecture are the problem list of Shing-Tung Yau, the books of Marcel Berger, as well as the books.
The Conjecture claims that any convex, closed and sufficiently smooth surface in three dimensional Euclidean space needs to admit at least two umbilic points. In the sense of the Conjecture, the spheroid with only two umbilic points and the sphere, all points of which are umbilic, are examples of surfaces with minimal and maximal numbers of umbilics. For the conjecture to be well posed, or the umbilic points to be well-defined, the surface needs to be at least twice differentiable.
Mathematical research on an approach by a local umbilic index estimate for real analytic surfaces
The invited address of Stefan Cohn-Vossen to the International Congress of Mathematicians of 1928 in Bologna was on the subject and in the 1929 edition of Wilhelm Blaschke's third volume on Differential Geometry  he states:
While this book goes into print, Mr Cohn-Vossen has succeeded in proving that closed real-analytic surfaces do not have umbilic points of index > 2 (invited talk at the ICM in Bologna 1928). This proves the conjecture of Carathéodory for such surfaces, namely that they need to have at least two umbilics.
Here Blaschke's index is twice the usual definition for index of an umbilic point, and the global conjecture follows by the Poincaré–Hopf index theorem. No paper was submitted by Cohn-Vossen to the proceedings of the International Congress, while in later editions of Blaschke's book the above comments were removed. It is therefore reasonable to assume that this work was inconclusive.
For analytic surfaces, an affirmative answer to this conjecture was given in 1940 by Hans Ludwig Hamburger in a long paper published in three parts. The approach of Hamburger was also via a local index estimate for isolated umbilics, which he had shown to imply the Conjecture in his earlier work. In 1943, a shorter proof was proposed by Gerrit Bol, see also, but, in 1959, Tilla Klotz  found and corrected a gap in Bol's proof in. Her proof, in turn, was announced to be incomplete in Hanspeter Scherbel's dissertation (no results of that dissertation related to the Carathéodory conjecture were published for decades, at least nothing was published up to June 2009). Among other publications we refer to papers.
All the proofs mentioned above are based on Hamburger's reduction of the Carathéodory conjecture to the following conjecture: the index of every isolated umbilic point is never greater than one. Roughly speaking, the main difficulty lies in resolution of singularities generated by umbilical points. All the above-mentioned authors resolve the singularities by induction on 'degree of degeneracy' of the umbilical point, but none of them was able to present the induction process clearly.
In 2002, Vladimir Ivanov revisited the work of Hamburger on analytic surfaces with the following stated intent:
"First, considering analytic surfaces, we assert with full responsibility that Carathéodory was right. Second, we know how this can be proved rigorously. Third, we intend to exhibit here a proof which, in our opinion, will convince every reader who is really ready to undertake a long and tiring journey with us."
First he follows the way passed by Gerrit Bol and Tilla Klotz, but later he proposes his own way for singularity resolution where crucial role belongs to complex analysis (more precisely, to techniques involving analytic implicit functions, Weierstrass preparation theorem, Puiseux series, and circular root systems).
Mathematical research on the original global conjecture for smooth surfaces
In 2008, Guilfoyle and Klingenberg announced a proof of the global conjecture for surfaces of smoothness C3,\alpha. Their method uses neutral Kähler geometry of the Klein quadric, Mean curvature flow, the Riemann–Roch index theorem, and the Sard-Smale Theorem on regular values of Fredholm operators. This preprint has not yet been published, however.
In 2012, Ghomi and Howard showed, using a Möbius transformation, that the global conjecture for surfaces of smoothness C2 can be reformulated in terms of the number of umbilic points on graphs subject to certain asymptotics of the gradient.
- Sitzungsberichte der Berliner Mathematischen Gesellschaft, 210. Sitzung am 26. März 1924, Dieterichsche Universitätsbuchdruckerei, Göttingen 1924
- Einfache Bemerkungen über Nabelpunktskurven, in: Festschrift 25 Jahre Technische Hochschule Breslau zur Feier ihres 25jährigen Bestehens, 1910—1935, Verlag W. G. Korn, Breslau, 1935, pp 105 - 107, and in: Constantin Carathéodory, Gesammelte Mathematische Schriften, Verlag C. H. Beck, München, 1957, vol 5, 26–30
- A mathematician's miscellany, Nabu Press (August 31, 2011) ISBN 978-1179121512
- H. Hamburger, Beweis einer Caratheodoryschen Vermutung. I, Ann. Math. (2) 41, 63—86 (1940); Beweis einer Caratheodoryschen Vermutung. II, Acta Math. 73, 175—228 (1941), and Beweis einer Caratheodoryschen Vermutung. III, Acta Math. 73, 229—332 (1941)
- D. J. Struik, Differential Geometry in the large, Bull. Amer. Math. Soc. 37, No 2, 49—62 (1931). doi:10.1090/S0002-9904-1931-05094-1
- S. T. Yau, Problem Section p. 684, in: Seminar on Differential Geometry, ed. S.T. Yau, Annals of Mathematics Studies 102, Princeton 1982
- M. Berger, A Panoramic View of Riemannian Geometry, Springer 2003 ISBN 3-540-65317-1
- M. Berger,Geometry Revealed: A Jacob's Ladder to Modern Higher Geometry , Springer 2010 ISBN 3-540-70996-7
- I. Nikolaev, Foliations on Surfaces , Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge A, Series of Modern Surveys in Mathematics, Springer 2001 ISBN 3-540-67524-8
- D. J. Struik, Lectures on Classical Differential Geometry, Dover 1978 ISBN 0-486-65609-8
- V. A. Toponogov, Differential Geometry of Curves and Surfaces: A Concise Guide, Birkhäuser, Boston 2006 ISBN 978-0-8176-4402-4
- R.V. Gamkrelidze (Ed.), Geometry I: Basic Ideas and Concepts of Differential Geometry , Encyclopaedia of Mathematical Sciences, Springer 1991 ISBN 0-387-51999-8
- S. Cohn-Vossen, Der Index eines Nabelpunktes im Netz der Krümmungslinien, Proceedings of the International Congress of Mathematicians, vol II, Nicola Zanichelli Editore, Bologna 1929
- W. Blaschke, Differentialgeometrie der Kreise und Kugeln, Vorlesungen über Differentialgeometrie, vol. 3, Grundlehren der mathematischen Wissenschaften XXIX, Springer-Verlag, Berlin 1929
- H. Hamburger 1922 Ein Satz über Kurvennetze auf geschlossenen Flächen, Sitzungsberichte der Preußischen Akademie der Wissenschaften zu Berlin 21, 258 - 262 (1922)
- H. Hamburger 1924 Über Kurvennetze mit isolierten Singularitäten auf geschossenen Flächen, Math. Z. 19, 50 - 66 (1924)
- G. Bol, Über Nabelpunkte auf einer Eifläche, Math. Z. 49, 389—410 (1944)
- W. Blaschke, Sugli ombelichi d'un ovaloide, Atti Convegno Mat. Roma 1942 (1942), pp. 201–208 (1945)
- On G. Bol's proof of Carathéodory's conjecture, Commun. Pure Appl. Math. 12, 277—311 (1959)
- H. Scherbel, A new proof of Hamburger's index theorem on umbilical points, Dissertation no. 10281 (1993), ETH Zürich
- C. J. Titus, A proof of a conjecture of Loewner and of the conjecture of Carathéodory on umbilic points, Acta Math. 131, No 1—2, 43—77 (1973)
- J. Sotomayor, L. F. Mello, A note on some developments on Carathéodory conjecture on umbilic points, Exposition Math. 17, No 1, 49—58 (1999). ISSN 0723-0869
- C. Gutierrez, J. Sotomayor, Lines of curvature, umbilic points and Carathéodory conjecture, Resen. Inst. Mat. Estat. Univ. São Paulo, 3, No 3, 291—322 (1998). ISSN 0104-3854
- V. V. Ivanov, The analytic Carathéodory conjecture, Sib. Math. J. 43, No. 2, 251—322 (2002). ISSN 0037-4474. doi:10.1023/A:1014797105633
- B. Guilfoyle and W. Klingenberg, "Proof of the Carathéodory Conjecture",  2013.
- M. Ghomi, "Open problems in geometry of curves and surfaces",  2017.
- M. Ghomi and R. Howard, Normal curvatures of asymptotically constant graphs and Carathéodory's conjecture Archived June 5, 2015, at the Wayback Machine., Proc. Amer. Math. Soc. 140 (2012), 4323-4335.