The method of moving frames originated in the early nineteenth century with the notion of the Frenet frame along a curve in Euclidean space. Later, Darboux expanded this idea to the study of surfaces. The method was brought to its full power in the early twentieth century by Elie Cartan, and its development continues today with the work of Fels, Olver, and others.

This book is an introduction to the method of moving frames as developed by Cartan, at a level suitable for beginning graduate students familiar with the geometry of curves and surfaces in Euclidean space. The main focus is on the use of this method to compute local geometric invariants for curves and surfaces in various 3-dimensional homogeneous spaces, including Euclidean, Minkowski, equi-affine, and projective spaces. Later chapters include applications to several classical problems in differential geometry, as well as an introduction to the nonhomogeneous case via moving frames on Riemannian manifolds.

The book is written in a reader-friendly style, building on already familiar concepts from curves and surfaces in Euclidean space. A special feature of this book is the inclusion of detailed guidance regarding the use of the computer algebra system Maple™ to perform many of the computations involved in the exercises.

An excellent and unique graduate level exposition of the differential geometry of curves, surfaces and higher-dimensional submanifolds of homogeneous spaces based on the powerful and elegant method of moving frames. The treatment is self-contained and illustrated through a large number of examples and exercises, augmented by Maple code to assist in both concrete calculations and plotting. Highly recommended.

—Niky Kamran, McGill University

The method of moving frames has seen a tremendous explosion of research activity in recent years, expanding into many new areas of applications, from computer vision to the calculus of variations to geometric partial differential equations to geometric numerical integration schemes to classical invariant theory to integrable systems to infinite-dimensional Lie pseudo-groups and beyond. Cartan theory remains a touchstone in modern differential geometry, and Clelland's book provides a fine new introduction that includes both classic and contemporary geometric developments and is supplemented by Maple symbolic software routines that enable the reader to both tackle the exercises and delve further into this fascinating and important field of contemporary mathematics.

Recommended for students and researchers wishing to expand their geometric horizons.

—Peter Olver, University of Minnesota

Readership

Undergraduate and graduate students interested in differential geometry.

• Juan A. Aledo, José M. Espinar, and José A. Gálvez, Timelike surfaces in the Lorentz-Minkowski space with prescribed Gaussian curvature and Gauss map, J. Geom. Phys. 56 (2006), no. 8, 1357–1369.
• Robert L. Anderson and Nail H. Ibragimov, Lie-Bäcklund Transformations in Applications, SIAM Studies in Applied Mathematics, vol. 1, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1979.
• M. A. Akivis and B. A. Rosenfeld, Élie Cartan (1869–1951), Translations of Mathematical Monographs, vol. 123, American Mathematical Society, Providence, RI, 1993, translated from the Russian manuscript by V. V. Goldberg.
• A. V. Bäcklund, Om ytor med konstant negativ krökning, Lunds Universitets Årsskrift 19 (1883), 1–48.
• R. L. Bryant, S. S. Chern, R. B. Gardner, H. L. Goldschmidt, and P. A. Griffiths, Exterior Differential Systems, Mathematical Sciences Research Institute Publications, vol. 18, Springer-Verlag, New York, 1991.
• Richard L. Bishop and Samuel I. Goldberg, Tensor Analysis on Manifolds, Dover Publications Inc., New York, 1980, corrected reprint of the 1968 original.
• Luigi Bianchi, Ricerche sulle superficie elicoidali e sulle superficie a curvatura costante, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 2 (1879), 285–341.
• Wilhelm Blaschke, Vorlesungen über Differentialgeometrie, vol. II, Springer, Berlin, 1923.
• —, Gesammelte Werke. Band 4, Thales-Verlag, Essen, 1985, Affine Differentialgeometrie. Differentialgeometrie der Kreis- und Kugelgruppen. [Affine differential geometry. Differential geometry of circle and ball groups], with commentaries by Werner Burau and Udo Simon, edited by Burau, S. S. Chern, K. Leichtweiß, H. R. Müller, L. A. Santaló, Simon and K. Strubecker.
• James J. Callahan, The Geometry of Spacetime: An Introduction to Special and General Relativity, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 2000.
• Élie Cartan, Sur la structure des groupes infinis de transformation, Ann. Sci. École Norm. Sup. (3) 21 (1904), 153–206.
• —, Sur la déformation projective des surfaces, Ann. Sci. École Norm. Sup. (3) 37 (1920), 259–356.
• —, Sur la possibilité de plonger un espace riemannien donné dans un espace euclidéen, Ann. Soc. Polon. Math. 6 (1927), 1–7.
• —, La théorie des groupes finis et continus et l’analysis situs, Mémorial des sciences mathématiques, no. 42, Gauthier-Villars et Cie, 1930.
• —, La Méthode du Repère Mobile, la Théorie des Groupes Continus, et les Espaces Généralisés, Exposés de Géométrie, no. 5, Hermann, Paris, 1935.
• —, Leçons sur la Géométrie des Espaces de Riemann, 2nd ed., Gauthier-Villars, Paris, 1946.
• —, Leçons sur la Géométrie Projective Complexe. La Théorie des Groupes Finis et Continus et la Géométrie Différentielle Traitées Par la Méthode du Repère Mobile. Leçons sur la Théorie des Espaces à Connexion Projective, Les Grands Classiques Gauthier-Villars. [Gauthier-Villars Great Classics], Éditions Jacques Gabay, Sceaux, 1992, reprint of the 1931, 1937, and 1937 editions.
• Jeanne Clelland, Edward Estrada, Molly May, Jonah Miller, and Sean Peneyra, A Tale of Two Arc Lengths: Metric Notions for Curves in Surfaces in Equiaffine Space, Proc. Amer. Math. Soc. 142 (2014), 2543–2558.
• Shiing-shen Chern, On integral geometry in Klein spaces, Ann. of Math. (2) 43 (1942), 178–189.
• Jeanne N. Clelland, Lie groups and the method of moving frames, 1999, lecture videos from Mathematical Sciences Research Institute Summer Graduate Workshop, http://www.msri.org/publications/video/index2.html.
• —, Totally quasi-umbilic timelike surfaces in $\mathbb {R}^{1,2}$, Asian J. Math. 16 (2012), 189–208.
• Shiing Shen Chern and Chuu Lian Terng, An analogue of Bäcklund’s theorem in affine geometry, Rocky Mountain J. Math. 10 (1980), no. 1, 105–124.
• Shiing-shen Chern and Keti Tenenblat, Pseudo-spherical surfaces and evolution equations, Stud. Appl. Math. 74 (1986), 55–83.
• Bernard Dacorogna, Introduction to the Calculus of Variations, 2nd ed., Imperial College Press, London, 2008.
• Gaston Darboux, Leçons sur la Théorie Générale des Surfaces et les Applications Géométriques du Calcul Infinitésimal. Deuxième partie, Chelsea Publishing Co., Bronx, N.Y., 1972, Les Congruences et les Équations Linéaires aux Dérivées Partielles. Les Lignes Tracées sur les Surfaces, réimpression de la deuxième édition de 1915.
• —, Leçons sur la Théorie Générale des Surfaces et les Applications Géométriques du Calcul Infinitésimal. Première partie, Chelsea Publishing Co., Bronx, N.Y., 1972, Généralités. Coordonnées curvilignes. Surfaces minima, réimpression de la deuxième édition de 1914.
• —, Leçons sur la Théorie Générale des Surfaces et les Applications Géométriques du Calcul Infinitésimal. Quatrième partie, Chelsea Publishing Co., Bronx, N.Y., 1972, Déformation Infiniment Petite et Representation Sphérique, réimpression de la première édition de 1896.
• —, Leçons sur la Théorie Générale des Surfaces et les Applications Géométriques du Calcul Infinitésimal. Troisième partie, Chelsea Publishing Co., Bronx, N.Y., 1972, Lignes Géodésiques et Courbure Géodésique. Paramètres Différentiels. Déformation des Surfaces, réimpression de la première édition de 1894.
• Manfredo P. do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall Inc., Englewood Cliffs, N.J., 1976, translated from the Portuguese.
• —, Riemannian Geometry, Mathematics: Theory & Applications, Birkhäuser Boston Inc., Boston, MA, 1992, translated from the second Portuguese edition by Francis Flaherty.
• B. A. Dubrovin, A. T. Fomenko, and S. P. Novikov, Modern Geometry—Methods and Applications. Part I: The Geometry of Surfaces, Transformation Groups, and Fields, 2nd ed., Graduate Texts in Mathematics, vol. 93, Springer-Verlag, New York, 1992, translated from the Russian by Robert G. Burns.
• Jesse Douglas, Solution of the problem of Plateau, Trans. Amer. Math. Soc. 33 (1931), no. 1, 263–321.
• Luther Pfahler Eisenhart, A Treatise on the Differential Geometry of Curves and Surfaces, Dover Publications Inc., New York, 1960.
• J. Favard, Cours de géométrie différentielle locale, Gauthier-Villars, Paris, 1957.
• Mark Fels and Peter J. Olver, Moving coframes. I. A practical algorithm, Acta Appl. Math. 51 (1998), no. 2, 161–213.
• —, Moving coframes. II. Regularization and theoretical foundations, Acta Appl. Math. 55 (1999), no. 2, 127–208.
• A. T. Fomenko, The Plateau Problem. Part I, Studies in the Development of Modern Mathematics, vol. 1, Gordon and Breach Science Publishers, New York, 1990, Historical survey, translated from the Russian.
• J. F. Frenet, Sur les Courbes à Double Courbure, Ph.D. thesis, Toulouse, 1847.
• José A. Gálvez, Surfaces of constant curvature in $3$-dimensional space forms, Mat. Contemp. 37 (2009), 1–42.
• Robert B. Gardner, The Method of Equivalence and Its Applications, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 58, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989.
• Mark L. Green, The moving frame, differential invariants and rigidity theorems for curves in homogeneous spaces, Duke Math. J. 45 (1978), no. 4, 735–779.
• P. Griffiths, On Cartan’s method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry, Duke Math. J. 41 (1974), 775–814.
• Heinrich W. Guggenheimer, Differential Geometry, Dover Publications, Inc., New York, 1977, corrected reprint of the 1963 edition, Dover Books on Advanced Mathematics.
• D. Hilbert and S. Cohn-Vossen, Geometry and the Imagination, Chelsea Publishing Company, New York, N.Y., 1952, translated by P. Neményi.
• Qing Han and Jia-Xing Hong, Isometric Embedding of Riemannian Manifolds in Euclidean Spaces, Mathematical Surveys and Monographs, vol. 130, American Mathematical Society, Providence, RI, 2006.
• Thomas A. Ivey and J. M. Landsberg, Cartan for Beginners: Differential Geometry via Moving Frames and Exterior Differential Systems, Graduate Studies in Mathematics, vol. 61, American Mathematical Society, Providence, RI, 2003.
• M. Janet, Sur la possibilité de plonger un espace Riemannien donné dans un espace euclidien, Ann. Soc. Polon. Math. 5 (1926), 38–43.
• Gary R. Jensen, Higher Order Contact of Submanifolds of Homogeneous Spaces, Springer-Verlag, Berlin, 1977, Lecture Notes in Mathematics, Vol. 610.
• Felix Klein, A comparative review of recent researches in geometry, Bull. Amer. Math. Soc. 2 (1893), no. 10, 215–249.
• —, Vergleichende Betrachtungen über neuere geometrische Forschungen, Math. Ann. 43 (1893), no. 1, 63–100.
• Shoshichi Kobayashi, Transformation Groups in Differential Geometry, Classics in Mathematics, Springer-Verlag, Berlin, 1995, reprint of the 1972 edition.
• Irina Kogan, Two algorithms for a moving frame construction, Canad. J. Math. 55 (2003), 266–291.
• John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Graduate Texts in Mathematics, vol. 218, Springer, New York, 2013.
• Elizabeth Louise Mansfield, A Practical Guide to the Invariant Calculus, Cambridge Monographs on Applied and Computational Mathematics, vol. 26, Cambridge University Press, Cambridge, 2010.
• Tilla Klotz Milnor, Harmonic maps and classical surface theory in Minkowski $3$-space, Trans. Amer. Math. Soc. 280 (1983), no. 1, 161–185.
• Hermann Minkowski, Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (1907/8), 53–111.
• —, Raum und Zeit, Jahresbericht der Deutschen Mathematiker-Vereinigung (1908/9), 75–88.
• Sebastián Montiel and Antonio Ros, Curves and surfaces, Graduate Studies in Mathematics, vol. 69, American Mathematical Society, Providence, RI; Real Sociedad Matemática Española, Madrid, 2005, translated and updated from the 1998 Spanish edition by the authors.
• John Nash, The imbedding problem for Riemannian manifolds, Ann. of Math. (2) 63 (1956), 20–63.
• Mehdi Nadjafikhah and Ali Mahdipour Sh., Affine classification of $n$-curves, Balkan J. Geom. Appl. 13 (2008), no. 2, 66–73.
• Katsumi Nomizu and Takeshi Sasaki, Affine Differential Geometry, Geometry of affine immersions, Cambridge Tracts in Mathematics, vol. 111, Cambridge University Press, Cambridge, 1994.
• Peter Olver, Applications of Lie Groups to Differential Equations, 2nd ed., Graduate Texts in Mathematics, Springer-Verlag, New York, 2000.
• —, Recent advances in the theory and application of Lie pseudo-groups, XVIII International Fall Workshop on Geometry and Physics, AIP Conf. Proc., vol. 1260, Amer. Inst. Phys., Melville, NY, 2010, pp. 35–63.
• —, Lectures on Moving Frames, Symmetries and Integrability of Difference Equations (Decio Levi, Peter Olver, Zora Thomova, and Pavel Winternitz, eds.), London Mathematical Society Lecture Note Series, vol. 381, Cambridge University Press, Cambridge, 2011, Lectures from the Summer School (Séminaire de Máthematiques Supérieures) held at the Université de Montréal, Montréal, QC, June 8–21, 2008.
• —, Recursive moving frames, Results Math. 60 (2011), 423–452.
• Barrett O’Neill, Semi-Riemannian Geometry, with applications to relativity, Pure and Applied Mathematics, vol. 103, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1983.
• —, Elementary Differential Geometry, 2nd ed., Elsevier/Academic Press, Amsterdam, 2006.
• John Oprea, Differential Geometry and Its Applications, 2nd ed., Classroom Resource Materials Series, Mathematical Association of America, Washington, DC, 2007.
• V. Ovsienko and S. Tabachnikov, Projective Differential Geometry Old and New, Cambridge Tracts in Mathematics, vol. 165, Cambridge University Press, Cambridge, 2005.
• Tibor Radó, On Plateau’s problem, Ann. of Math. (2) 31 (1930), no. 3, 457–469.
• E. G. Reyes, Pseudo-spherical surfaces and integrability of evolution equations, J. Diff. Eq. 147 (1998), 195–230.
• C. Rogers and W. F. Shadwick, Bäcklund Transformations and Their Applications, Mathematics in Science and Engineering, vol. 161, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1982.
• C. Rogers and W. K. Schief, Bäcklund and Darboux Transformations, Geometry and modern applications in soliton theory, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002.
• C. E. Senff, Theoremata Princioalia e Theoria Curvarum et Su Perficierum, Dorpat Univ., 1831.
• J. Serret, Mémoire sur quelques formules relatives à la théorie des courbes à double courbure, J. Math. Pures Appl. 16 (1851), 193–207.
• Mary D. Shepherd, Line congruences as surfaces in the space of lines, Differential Geom. Appl. 10 (1999), no. 1, 1–26.
• Michael Spivak, A Comprehensive Introduction to Differential Geometry. 5 Vols., 2nd ed., Publish or Perish Inc., Wilmington, DE, 1979.
• James Stewart, Calculus: Early Transcendentals, 6th ed., Stewart’s Calculus Series, Brooks Cole, Belmont, CA, 2007.
• Bu Chin Su, Affine Differential Geometry, Science Press, Beijing, 1983.
• Kazumi Tsukada, Totally geodesic submanifolds of Riemannian manifolds and curvature-invariant subspaces, Kodai Math. J. 19 (1996), no. 3, 395–437.
• K. Weierstrass, Über die Flächen deren mittlere Krümmung überall gleich null ist., Monatsber. Berliner Akad. (1866), 612–625, 855–856.
• T. J. Willmore, The definition of Lie derivative, Proc. Edinburgh Math. Soc. (2) 12 (1960/1961), 27–29.

• Juan A. Aledo, José M. Espinar, and José A. Gálvez, Timelike surfaces in the Lorentz-Minkowski space with prescribed Gaussian curvature and Gauss map, J. Geom. Phys. 56 (2006), no. 8, 1357–1369.
• Robert L. Anderson and Nail H. Ibragimov, Lie-Bäcklund Transformations in Applications, SIAM Studies in Applied Mathematics, vol. 1, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1979.
• M. A. Akivis and B. A. Rosenfeld, Élie Cartan (1869–1951), Translations of Mathematical Monographs, vol. 123, American Mathematical Society, Providence, RI, 1993, translated from the Russian manuscript by V. V. Goldberg.
• A. V. Bäcklund, Om ytor med konstant negativ krökning, Lunds Universitets Årsskrift 19 (1883), 1–48.
• R. L. Bryant, S. S. Chern, R. B. Gardner, H. L. Goldschmidt, and P. A. Griffiths, Exterior Differential Systems, Mathematical Sciences Research Institute Publications, vol. 18, Springer-Verlag, New York, 1991.
• Richard L. Bishop and Samuel I. Goldberg, Tensor Analysis on Manifolds, Dover Publications Inc., New York, 1980, corrected reprint of the 1968 original.
• Luigi Bianchi, Ricerche sulle superficie elicoidali e sulle superficie a curvatura costante, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 2 (1879), 285–341.
• Wilhelm Blaschke, Vorlesungen über Differentialgeometrie, vol. II, Springer, Berlin, 1923.
• —, Gesammelte Werke. Band 4, Thales-Verlag, Essen, 1985, Affine Differentialgeometrie. Differentialgeometrie der Kreis- und Kugelgruppen. [Affine differential geometry. Differential geometry of circle and ball groups], with commentaries by Werner Burau and Udo Simon, edited by Burau, S. S. Chern, K. Leichtweiß, H. R. Müller, L. A. Santaló, Simon and K. Strubecker.
• James J. Callahan, The Geometry of Spacetime: An Introduction to Special and General Relativity, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 2000.
• Élie Cartan, Sur la structure des groupes infinis de transformation, Ann. Sci. École Norm. Sup. (3) 21 (1904), 153–206.
• —, Sur la déformation projective des surfaces, Ann. Sci. École Norm. Sup. (3) 37 (1920), 259–356.
• —, Sur la possibilité de plonger un espace riemannien donné dans un espace euclidéen, Ann. Soc. Polon. Math. 6 (1927), 1–7.
• —, La théorie des groupes finis et continus et l’analysis situs, Mémorial des sciences mathématiques, no. 42, Gauthier-Villars et Cie, 1930.
• —, La Méthode du Repère Mobile, la Théorie des Groupes Continus, et les Espaces Généralisés, Exposés de Géométrie, no. 5, Hermann, Paris, 1935.
• —, Leçons sur la Géométrie des Espaces de Riemann, 2nd ed., Gauthier-Villars, Paris, 1946.
• —, Leçons sur la Géométrie Projective Complexe. La Théorie des Groupes Finis et Continus et la Géométrie Différentielle Traitées Par la Méthode du Repère Mobile. Leçons sur la Théorie des Espaces à Connexion Projective, Les Grands Classiques Gauthier-Villars. [Gauthier-Villars Great Classics], Éditions Jacques Gabay, Sceaux, 1992, reprint of the 1931, 1937, and 1937 editions.
• Jeanne Clelland, Edward Estrada, Molly May, Jonah Miller, and Sean Peneyra, A Tale of Two Arc Lengths: Metric Notions for Curves in Surfaces in Equiaffine Space, Proc. Amer. Math. Soc. 142 (2014), 2543–2558.
• Shiing-shen Chern, On integral geometry in Klein spaces, Ann. of Math. (2) 43 (1942), 178–189.
• Jeanne N. Clelland, Lie groups and the method of moving frames, 1999, lecture videos from Mathematical Sciences Research Institute Summer Graduate Workshop, http://www.msri.org/publications/video/index2.html.
• —, Totally quasi-umbilic timelike surfaces in $\mathbb {R}^{1,2}$, Asian J. Math. 16 (2012), 189–208.
• Shiing Shen Chern and Chuu Lian Terng, An analogue of Bäcklund’s theorem in affine geometry, Rocky Mountain J. Math. 10 (1980), no. 1, 105–124.
• Shiing-shen Chern and Keti Tenenblat, Pseudo-spherical surfaces and evolution equations, Stud. Appl. Math. 74 (1986), 55–83.
• Bernard Dacorogna, Introduction to the Calculus of Variations, 2nd ed., Imperial College Press, London, 2008.
• Gaston Darboux, Leçons sur la Théorie Générale des Surfaces et les Applications Géométriques du Calcul Infinitésimal. Deuxième partie, Chelsea Publishing Co., Bronx, N.Y., 1972, Les Congruences et les Équations Linéaires aux Dérivées Partielles. Les Lignes Tracées sur les Surfaces, réimpression de la deuxième édition de 1915.
• —, Leçons sur la Théorie Générale des Surfaces et les Applications Géométriques du Calcul Infinitésimal. Première partie, Chelsea Publishing Co., Bronx, N.Y., 1972, Généralités. Coordonnées curvilignes. Surfaces minima, réimpression de la deuxième édition de 1914.
• —, Leçons sur la Théorie Générale des Surfaces et les Applications Géométriques du Calcul Infinitésimal. Quatrième partie, Chelsea Publishing Co., Bronx, N.Y., 1972, Déformation Infiniment Petite et Representation Sphérique, réimpression de la première édition de 1896.
• —, Leçons sur la Théorie Générale des Surfaces et les Applications Géométriques du Calcul Infinitésimal. Troisième partie, Chelsea Publishing Co., Bronx, N.Y., 1972, Lignes Géodésiques et Courbure Géodésique. Paramètres Différentiels. Déformation des Surfaces, réimpression de la première édition de 1894.
• Manfredo P. do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall Inc., Englewood Cliffs, N.J., 1976, translated from the Portuguese.
• —, Riemannian Geometry, Mathematics: Theory & Applications, Birkhäuser Boston Inc., Boston, MA, 1992, translated from the second Portuguese edition by Francis Flaherty.
• B. A. Dubrovin, A. T. Fomenko, and S. P. Novikov, Modern Geometry—Methods and Applications. Part I: The Geometry of Surfaces, Transformation Groups, and Fields, 2nd ed., Graduate Texts in Mathematics, vol. 93, Springer-Verlag, New York, 1992, translated from the Russian by Robert G. Burns.
• Jesse Douglas, Solution of the problem of Plateau, Trans. Amer. Math. Soc. 33 (1931), no. 1, 263–321.
• Luther Pfahler Eisenhart, A Treatise on the Differential Geometry of Curves and Surfaces, Dover Publications Inc., New York, 1960.
• J. Favard, Cours de géométrie différentielle locale, Gauthier-Villars, Paris, 1957.
• Mark Fels and Peter J. Olver, Moving coframes. I. A practical algorithm, Acta Appl. Math. 51 (1998), no. 2, 161–213.
• —, Moving coframes. II. Regularization and theoretical foundations, Acta Appl. Math. 55 (1999), no. 2, 127–208.
• A. T. Fomenko, The Plateau Problem. Part I, Studies in the Development of Modern Mathematics, vol. 1, Gordon and Breach Science Publishers, New York, 1990, Historical survey, translated from the Russian.
• J. F. Frenet, Sur les Courbes à Double Courbure, Ph.D. thesis, Toulouse, 1847.
• José A. Gálvez, Surfaces of constant curvature in $3$-dimensional space forms, Mat. Contemp. 37 (2009), 1–42.
• Robert B. Gardner, The Method of Equivalence and Its Applications, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 58, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989.
• Mark L. Green, The moving frame, differential invariants and rigidity theorems for curves in homogeneous spaces, Duke Math. J. 45 (1978), no. 4, 735–779.
• P. Griffiths, On Cartan’s method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry, Duke Math. J. 41 (1974), 775–814.
• Heinrich W. Guggenheimer, Differential Geometry, Dover Publications, Inc., New York, 1977, corrected reprint of the 1963 edition, Dover Books on Advanced Mathematics.
• D. Hilbert and S. Cohn-Vossen, Geometry and the Imagination, Chelsea Publishing Company, New York, N.Y., 1952, translated by P. Neményi.
• Qing Han and Jia-Xing Hong, Isometric Embedding of Riemannian Manifolds in Euclidean Spaces, Mathematical Surveys and Monographs, vol. 130, American Mathematical Society, Providence, RI, 2006.
• Thomas A. Ivey and J. M. Landsberg, Cartan for Beginners: Differential Geometry via Moving Frames and Exterior Differential Systems, Graduate Studies in Mathematics, vol. 61, American Mathematical Society, Providence, RI, 2003.
• M. Janet, Sur la possibilité de plonger un espace Riemannien donné dans un espace euclidien, Ann. Soc. Polon. Math. 5 (1926), 38–43.
• Gary R. Jensen, Higher Order Contact of Submanifolds of Homogeneous Spaces, Springer-Verlag, Berlin, 1977, Lecture Notes in Mathematics, Vol. 610.
• Felix Klein, A comparative review of recent researches in geometry, Bull. Amer. Math. Soc. 2 (1893), no. 10, 215–249.
• —, Vergleichende Betrachtungen über neuere geometrische Forschungen, Math. Ann. 43 (1893), no. 1, 63–100.
• Shoshichi Kobayashi, Transformation Groups in Differential Geometry, Classics in Mathematics, Springer-Verlag, Berlin, 1995, reprint of the 1972 edition.
• Irina Kogan, Two algorithms for a moving frame construction, Canad. J. Math. 55 (2003), 266–291.
• John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Graduate Texts in Mathematics, vol. 218, Springer, New York, 2013.
• Elizabeth Louise Mansfield, A Practical Guide to the Invariant Calculus, Cambridge Monographs on Applied and Computational Mathematics, vol. 26, Cambridge University Press, Cambridge, 2010.
• Tilla Klotz Milnor, Harmonic maps and classical surface theory in Minkowski $3$-space, Trans. Amer. Math. Soc. 280 (1983), no. 1, 161–185.
• Hermann Minkowski, Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (1907/8), 53–111.
• —, Raum und Zeit, Jahresbericht der Deutschen Mathematiker-Vereinigung (1908/9), 75–88.
• Sebastián Montiel and Antonio Ros, Curves and surfaces, Graduate Studies in Mathematics, vol. 69, American Mathematical Society, Providence, RI; Real Sociedad Matemática Española, Madrid, 2005, translated and updated from the 1998 Spanish edition by the authors.
• John Nash, The imbedding problem for Riemannian manifolds, Ann. of Math. (2) 63 (1956), 20–63.
• Mehdi Nadjafikhah and Ali Mahdipour Sh., Affine classification of $n$-curves, Balkan J. Geom. Appl. 13 (2008), no. 2, 66–73.
• Katsumi Nomizu and Takeshi Sasaki, Affine Differential Geometry, Geometry of affine immersions, Cambridge Tracts in Mathematics, vol. 111, Cambridge University Press, Cambridge, 1994.
• Peter Olver, Applications of Lie Groups to Differential Equations, 2nd ed., Graduate Texts in Mathematics, Springer-Verlag, New York, 2000.
• —, Recent advances in the theory and application of Lie pseudo-groups, XVIII International Fall Workshop on Geometry and Physics, AIP Conf. Proc., vol. 1260, Amer. Inst. Phys., Melville, NY, 2010, pp. 35–63.
• —, Lectures on Moving Frames, Symmetries and Integrability of Difference Equations (Decio Levi, Peter Olver, Zora Thomova, and Pavel Winternitz, eds.), London Mathematical Society Lecture Note Series, vol. 381, Cambridge University Press, Cambridge, 2011, Lectures from the Summer School (Séminaire de Máthematiques Supérieures) held at the Université de Montréal, Montréal, QC, June 8–21, 2008.
• —, Recursive moving frames, Results Math. 60 (2011), 423–452.
• Barrett O’Neill, Semi-Riemannian Geometry, with applications to relativity, Pure and Applied Mathematics, vol. 103, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1983.
• —, Elementary Differential Geometry, 2nd ed., Elsevier/Academic Press, Amsterdam, 2006.
• John Oprea, Differential Geometry and Its Applications, 2nd ed., Classroom Resource Materials Series, Mathematical Association of America, Washington, DC, 2007.
• V. Ovsienko and S. Tabachnikov, Projective Differential Geometry Old and New, Cambridge Tracts in Mathematics, vol. 165, Cambridge University Press, Cambridge, 2005.
• Tibor Radó, On Plateau’s problem, Ann. of Math. (2) 31 (1930), no. 3, 457–469.
• E. G. Reyes, Pseudo-spherical surfaces and integrability of evolution equations, J. Diff. Eq. 147 (1998), 195–230.
• C. Rogers and W. F. Shadwick, Bäcklund Transformations and Their Applications, Mathematics in Science and Engineering, vol. 161, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1982.
• C. Rogers and W. K. Schief, Bäcklund and Darboux Transformations, Geometry and modern applications in soliton theory, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002.
• C. E. Senff, Theoremata Princioalia e Theoria Curvarum et Su Perficierum, Dorpat Univ., 1831.
• J. Serret, Mémoire sur quelques formules relatives à la théorie des courbes à double courbure, J. Math. Pures Appl. 16 (1851), 193–207.
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