Geometry

_____________________________________________________________________________________________________________________________________

**PERSONAL RESEARCH**

_____________________________________________________________________________________________________________________________________

__Published articles__

D. H. Delphenich, “Projective geometry and special relativity,” Ann. Phys. (Leipzig)

**15**(2006), 216-246

D. H. Delphenich, “A more direct representation for complex relativity,” Ann. Phys. (Leipzig)

**16**(2007) 615-639

__Invited talk at University of Missouri – Columbia__

D. H. Delphenich, “Projective geometry and spacetime structure”

__arXiv.org uploads__

D. H. Delphenich, "Singular Teleparallelism," arXiv:1809.02663

D. H. Delphenich, "Teleparallelism as anholonomic geometry," arXiv:1808.09318

D. H. Delphenich, "On the local integrability of almost-product structures defined by space-time metrics," arXiv:1607.03839

D. H. Delphenich, "Line geometry and electromagnetism III: groups of transformations," arXiv:1404.4330

D. H. Delphenich, "Line geometry and electromagnetism II: wave motion," arXiv:1311.6766

D. H. Delphenich, "Line geometry and electromagnetism I: basic structures," http://arxiv.org/abs/1309.2933

D. H. Delphenich, “Transverse geometry and physical observers,” arXiv:0711.2033

D. H. Delphenich, “Groups of motions and mechanics I: point mechanics,” arXiv:0708.1572

D. H. Delphenich, “Nonlinear connections and 1+3 splittings of spacetime, ” arXiv:gr-qc/0702115

D. H. Delphenich, “Spacetime G-structures II: geometry of the ground states,” arXiv:gr-qc/0401089

D. H. Delphenich, “Some Mathematical Issues Pertaining to Translational Gauge Theories,” arXiv:gr-qc/0401066

D. H. Delphenich, “Spacetime G-structures I: Topological Defects,” arXiv:gr-qc/0304016

D. H. Delphenich, “Proper Time Foliations of Lorentz Manifolds,” arXiv:gr-qc/0211066

D. H. Delphenich, “The Geometric Origin of the Madelung Potential,” arXiv:gr-qc/0211065

_______________________________________________________________________________________________________________________________________________________

**TRANSLATIONS**

_______________________________________________________________________________________________________________________________________________________

__Lie groups, pseudogroups, and groupoids__

W. Killing, "The composition of continuous, finite, transformation groups," Math. Ann.

**31**(1869), 252-290.

F. Engel, “On the defining equations of continuous transformation groups,” Math. Ann.

**27**(1886), 1-57.

S. Lie, “The foundations of the theory of infinite continuous groups - I” Leipz. Ber., issue III (1891), 316; Ges. Abh., Bd.

**6**, art. XI, pp. 300-330.

S. Lie, “The foundations of the theory of infinite continuous groups - II” Leipz. Ber., issue III (1891), 353-393; Ges. Abh., Bd.

**6**, art. XII, pp. 331-364.

P. Medolaghi, “On the theory of infinite continuous groups,” Annali di Matematica

**25**(1897), 179-217.

E. Vessiot, "On the theory of continuous groups," Ann. sci. de l'E.N.S. (3)

**20**(1903), 411-451.

U. Amaldi, "On the principal results obtained in the theory of transformation groups since the death of Sophus Lie (1898-1907)," Congres de la Societe Italienne pour le progres des Sciences, Parma, 1907.

E. Cartan, "The method of moving frames, the theory of continuous groups generalized spaces," Exposes de Geometrie, t. V, Hermann, Paris, 1935.

E. Cartan, “The structure of infinite groups,” Séminaire de Mathématiques, 4th year, 1926-1937; Oeuvres Complètes, v. ?, 1336-1384.

E. Cartan, "The theory of finite, continuous groups and analysis situs," Mem. Sc. math., Acad. Sci. Paris, fasc. 42, Gauthier-Villars, Paris, 1952.

S. S. Chern, “Infinite continuous groups,” in

*Géométrie différentielle*, V. LII, Colloque Internationale du C. N. R. S., Strasbourg, 1953.

C. Ehresmann, “Introduction to the theory of infinitesimal structures and Lie pseudogroups,” Coll. Intl. de Géom. diff. du C.N.R.S, Strasbourg (1953), 97-110.

Y. Matsushima, “Transitive Lie pseudogroups,” Séminaire Boubaki, May, 1955.

D. Bernard, "On the differential geometry of G-structures," Ann. Inst. Fourier

**10**(1960), 151-270.

Ngô Van Que, “On the prolongation of fiber bundles and infinitesimal structures,” Ann. Inst. Fourier, Grenoble

**17**, 1 (1967), 157-223.

__Projective, line, conformal, and algebraic geometry__

Ch. Dupin, "On the paths that are followed by light and elastic bodies in general under the phenomena of reflection and refraction," from

*Applications de geometrie et de mecanique...,*Courcier, Paris, 1822.

M. Gergonne, "Optics. Purely geometric proof of a fundamental principle in the theory of caustics," Ann. de Math. pures et appl.

**16**(1825-1826), 307-314.

A. F. Moebius, "On a special kind of dual relationship between figures in space," J. f. d. reine u. angew. Math.

**10**(1833), 317-341.

J. Plucker, "Discussion of the general form for light waves," J. f. reine u. angew. Math.

**19**(1839), 1-44.

J. Bertrand, "Memoir on the theory of surfaces," J. Math. pures et appl.

**9**(1844), 133-154.

J. Bertrand, "Note on the wave surface," Comptes rendus

**47**(1858), 817-819.

F. Brioschi, "On the lines of curvature of the wave surface," Ann. mat. pura ed appl.

**2**(1859), 135-136.

E. Combescure, "On the lines of curvature of the wave surface," Ann. di. mat. pura ed. appl.

**2**(1859), 278-285.

O. Bonnet, "Memoir on the theory of surfaces that can be mapped to a given surface," J. l’École poly.

**41**(1860), 209-230.

E. Kummer, "General theory of rectilinear ray systems," J. f. reine u. angew. Math.

**57**(1860), 189-230.

E. Kummer, (no title: rectilinear systems of light rays), Monatsber. d. konig.-preuss. Akad. d. Wiss. zu Berlin, phys-math. Klasse, 30 Juli 1860, pp. 469-474.

E. E. Kummer, "Three models of general, infinitely-thin, rectilinear ray sheaves that are composed of filaments, Monats. d. Konig. Preuss. Akad. Wiss. Berlin (1860), 469-474.

E. E. Kummer, "On atmospheric ray refraction," J. f. d. reine u. angew. Math.

**61**(1863), 263-275.

E. E. Kummer, "On the fourth-degree surfaces with sixteen singular points," Monats. d. Konig. Preuss. Akad. Wiss. Berlin (1864), 246-270.

E. E. Kummer, "On the ray systems whose focal surfaces are surfaces of degree four with sixteen singular points," Monats. d. Konig. Preuss. Akad. Wiss. Berlin (1864), 495-499.

R. Meibauer, "Theory of rectilinear systems of light rays," C. Luderitz, Berlin, 1864.

E. Beltrami, "On the flexion of ruled surfaces," Ann. Mat. pura ed appl. (1)

**7**(1863), 105-138.

E. E. Kummer, "On algebraic ray systems; in particular, on those of orders one and two," Abh. d. Kon. Akad. Wiss. zu Berlin (1866), 1-120.

J. Pluecker,

*A New Geometry of Space*, Teubner, Leipzig, 1868.

A. Clebsch, "On Pluecker complexes," Math. Ann.

**2**(1869), 1-8.

F. Klein, “On the theory of line complexes of first and second order,” Math. Ann.

**2**(1870); Ges. math. Abh., art. II.

F. Klein, “On line geometry and metric geometry,” Ges. math. Abh., art. VIII, pp. 106-126.

F. Klein, “On a geometric interpretation of the resolvents of algebraic equations,” Math. Ann..

**4**(1871); Ges. math. Abh., art. L, pp. 106-126.

S. Lie, “On complexes – in particular, line and sphere complexes – with applications to the theory of partial differential equations,” Math. Ann.

**5**(1872), 145-208.

F. Klein, “On certain differential equations that appear in line geometry,” Math. Ann.

**5**(1872); Ges. math. Abh., Art. IX.

M. Pasch, "On the focal surfaces of ray systems and the singularity surfaces of complexes," J. f. reine u. angew. Math.

**76**(1873), 156-169.

A. Voss, "Geometrical interpretation of the differential equation

*P dx*+

*Q dy*+

*R dz*= 0," Math. Ann.

**16**(1880), 556-559.

C. Geiser, "On a fundamental theorem in the kinematic geometry of space," J. f. d. reine u. angew. Math.

**90**(1881), 39-43.

A. Voss, "On the theory of general point-plane systems," Math. Ann.

**23**(1884), 45-81.

M. Blasendorff, "On the relations between two general ray systems...," Inaugural Dissertation, Berlin, 1883.

M. Blasendorff, "On optical ray systems," J. f. d. reine u. angew. Math.

**97**(1884), 172-176.

F. Klein, "On the transformation of the general second-degree equation in line coordinates into canonical coordinates," Math. Ann.

**23**(1884), 539-578.

C. Segre, " On the lines that have given moments with respect to fixed lines," J. f. d. reine u. angewandte Mathematik

**97**(1884), 95-110.

S. Lie and F. Klein, "On the principal tangent curves of the fourth-degree Kummer surface with 16 nodes," Math. Ann.

**23**(1884); F. Klein,

*Ges. math. Werke*, art. VI.

G. Hauck, "On the relationship between null systems and linear ray complexes and the polar systems of paraboloids of rotation," Zeit. f. Math. u. Phys.

**31**(1886), 362-368.

G. Darboux, "On the wave surface," Ann.sci. de l'E.N.S (3)

**6**(1889), 379-388.

E. Waelsch, "On the invariant theory of line geometry," Math. Ann.

**37**(1890), 141-152.

E. Study, "On systems of complex numbers and their application to the theory of transformation groups," Monatsber. f. Math. u. Phys.

**1**(1890), 283-354.

E. Mueller, "Line geometry, according to the principles of Grassmann's theory of extensions," Monats. f. Math. u. Phys.

**2**(1891), 267-290.

E. Cesaro,

*Lessons on Intrinsic Geometry,*Printed by the author-editor, Naples, 1896.

In two files: Chap. 1-XIV, Chap. XV-end

C. Burali-Forti, "The Grassmann method in projective geometry," Rend. Circ. matem. Palermo, Note I,

**10**(1896), 177-195; Note II,

**11**(1897), 64-84; Note III,

**15**(1901), 310-320.

G. Koenigs,

*Line Geometry, and its applications*, Gauthier-Villars, Paris, 1898.

S. Lie, "On the geometry of a Monge equation," Ber. Verh. Kon.Ges. Wiss. Leipzig

**50**(1898), 1-3.

G. Fubini, "Clifford parallelism in elliptic spaces," Laurea thesis, Pisa, 1900.

E. Study, "A New Branch of Geometry," Jahresberichte d. deutschen Math-Ver. XI, Heft 3 (1901), 97-123.

E. Study, "On non-Euclidian and line geometry," Jahresberichte d. deutschen Math-Ver. XI, Heft 8 and 9 (1901), 313-342.

K. Zindler,

*Line geometry, with applications,*v. I, Goschen, Leipzig, 1902.

E. v. Weber, "Complex motions," Ber. sachs. Ges. der Wiss (1903), 384-408.

J. Grunwald, "On dual numbers and their application in geometry," Monatsh. f. Math. u. Phys.

**17**(1906), 81-136.

K. Zindler, "The development and present state of differential line geometry," Jahres. der deutschen Math.-Verein.

**15**(1906), 185-213.

C. Segre, "Monge and line congruences," Biblioteca Math. (3)

**8**(1907), 321-324.

G. Sannia, "A sampling of the differential geometry of line complexes," Ann. di Mat. pura ed appl. (3)

**17**(1910), 179-223.

R. Weitzenbock, "On the system of forms of linear complexes in R3," Jber. d. Deut. Math.-Verein.

**19**(1910), 225-238.

H. Schlusser,

*The focal theory of linear ray congruences*, Hoffmann, Leipzig, 1912.

G. Fubini, "Fundamentals of the projective differential geometry of complexes and congruences of lines, I" Rend. Reale Accad. dei Lincei (5)

**27**(1918), 304-311.

G. Fubini, "Fundamentals of the projective differential geometry of complexes and congruences of lines, II" Rend. Reale Accad. dei Lincei (5)

**28**(1919), 32-39.

E. Vessiot,

*Lessons on Higher Geometry*, Hermann and Co., Paris, 1919: Chap. 1-VII, Chap. VIII-end.

H. Beck, "On linear manifolds of somas," Math. Ann.

**81**(1920), 187-218.

H. Beck, "On improper somas," Math. Ann.

**87**(1920), 152-156.

G. Thomsen, "On a common way of treating various geometries," Math. Zeit.

**21**(1924), 254-285.

E. Study, "Simplified foundation for Lie's sphere geometry," Sitz. preuss.Akad. Wiss. Berlin. math.-phys. Klasse

**27**(1926), 360-380.

E. A. Weiss,

*Introduction to line geometry and kinematics,*Teubner, Leipzig, 1935.

R. Sauer,

*Projective Line Geometry*, De Gruyter, Berlin and Leipzig, 1937.

B. L. van der Waerden,

*Introduction to Algebraic Geometry*, Dover, New York, 1945.

H. Knorrer, "The Fresnel wave surface," Lecture four of

*Mathematische Miniaturen 3: Arithmetik und Geometrie*.

*Vier Vorlesungen*, Birkhauser, Basel, 1986.

__Differential geometry (affine and projective)__

J.H. Jellett, "On the surface whose mean curvature is constant," J. pure et appl. math.

**18**(1853), 163-167.

De la Gournerie, "Study of the curvature of surfaces," J. math. pure appl.

**20**(1855), 145-156.

D. Codazzi, "On the surfaces whose deformations preserve the lines of curvature," Ann. di sci. mat. fis.

**7**(1856), 410-416.

J. Weingarten, "On a class of mutually-developable surfaces," J. reine angew. Math.

**59**(1861), 382-393.

R. Hoppe, "On the representation of curves by curvature and torsion," J. reine angew. Math.

**60**(1862), 182-187.

U. Dini, "On the differential equation of the surfaces that can be mapped to a given surface," Giorn. di mat. de Battaglini

**2**(1864), 282-288.

E. Combescure, "On functional determinants and curvilinear coordinates," Ann. de la. Ec. Norm. Sup. (1)

**4**(1867), 93-131.

E. Beltrami, "On the general theory of differential parameters," Memorie della Accademia della scienze dell'Istituto Bologna (2)

**8**(1868), 549-590.

L. Kronecker, "On systems of functions of several variables," Monatsber. Kon. Akad. Wiss. (1869), 688-698.

F. Suvaroff, "On the characteristics of three-dimensional systems," Bull. Sci. math. et Astron.

**4**(1873), 172-192.

J. Weingarten, "On the theory of mutually-developable surfaces," Festschrift der Kon. Techn. Hochschule zu Berlin (1884), 1-43.

J. Weingarten, "On the infinitely-small deformations of a flexible, inextensible surface," Sitzungsber. Kön. Preuss. Akad. Wiss Berlin (1886), 83-91.

G. Darboux,

*Book I: Applications of the theory of relative motion to geometry*, from

*Lessons on the general theory of surfaces,*Part One, Gauthier-Villars, Paris, 1887.

G. Darboux,

*Book II: Various curvil*

*inear coordinate systems*, from

*Lessons on the general theory of surfaces, etc.*Part One, Gauthier-VIllars, Paris, 1887.

J. Weingarten, "On the deformation of a flexible, inextensible surface," J. fur reine u. angew. Math.

**100**(1887), 296-310.

G. Darboux, Chapter I: "General notions on congruences," from

*Lecons sur la theorie des surfaces, etc.*, Book IV, Gauthier-Villars, Paris, 1889.

G. Darboux, Chapter XIII: "Normal lines to a surface," from

*Lecons sur la theorie des surfaces, etc.*, Book IV, Gauthier-Villars, Paris, 1889.

G. Darboux,

*Book V:*

*Lines that are*

*traced on surfaces,*from

*Lessons on the general theory of surfaces, etc.*Part Two, Gauthier-VIllars, Paris, 1889.

H. Stahl and V. Kommerell,

*The Basic formulas for the General Theory of Surfaces,*Teubner, Leipzig, 1893.

G. Darboux, Note VIII: "On the asymptotic lines and lines of curvature of the Fresnel wave surface," from

*Lecons sur la theorie des surfaces*, Part IV, Gauthier-Villars, Paris, 1896, pp. 466-488.

E. Cesaro, "On the use of Grassmann numbers," from

*Lezioni di geometria intrinseca*, from the author-publisher, Naples, 1896.

C. Burali-Forti,

*Introduction to Differential Geometry, following the method of H. Grassmann*, Gauthier-Villars, Paris, 1897.

L. Bianchi,

*Chapter I, Curves of double curvature,*from vol. 1 of

*Lezioni di geometria differenziale*, 2nd ed., Enrico Spoerri, Pisa, 1902.

L. Bianchi, Chapter IV: "Fundamental formulas from the theory of surfaces," from

*Lezioni di Geometria differenziale*, 2nd ed., v. 1, Enrico Spoerri, Pisa, 1902.

L. Bianchi: Chapter X: "Systems of (infinity-squared) rays or rectilinear congruences," from

*Lezioni di geometria differenziale*, 2nd ed., v. I, Enrico Spoerri, Pisa, 1902.

G. Ricci, "On the continuous groups of motions in an arbitrary three-dimensional manifold," Mem. mat. fis. della Soc. It. (3)

**12**(1902), 69-92.

G. Ricci, "One the continuous groups of motions in hyperspaces," Rend. Reale Accad. dei Lincei, classe sci., fis., mat. e nat. (5)

**14**(1905), 487-491.

G. Sannia, "The Combescure transformation and other analogous ones for twisted curves," Rend. Circ. Mat. Palermo

**20**(1905), 83-92.

G. Sannia, "Infinitesimal deformations of inextensible curves and correspondence by orthogonality of elements," Rend. Circ. Mat. Palermo

**21**(1906), 229-256.

P. Ernst, "The general Mannheim curve," Monatsh. fur Math. u. Phys.

**23**(1912), 289-296.

P. Ernst, "The Mannheim curve of a space curve," Monatsh. fur Math. u. Phys.

**25**(1914), 337-340.

H. Weyl, “Pure infinitesimal geometry,” Math. Zeit.

**2**(1918), 284-411.

H. Weyl, “On infinitesimal geometry: relationship to projective and conformal concepts,” Gött. Nachr. (1921), 99-112.

E. Cartan, “Spaces with conformal connections,” Ann. Soc. Pol. Math.

**2**(1923), 171-221

E. Cartan, “On manifolds with projective connections,” Bull. Soc. Math. France

**52**(1924), 205-241.

D. van Dantzig, “Theory of projective connections on

*n*-dimensional spaces,” Math. Ann.

**106**(1932), 400-454.

D. van Dantzig, “On general projective differential geometry. I – Relationship to affine geometry,” Proc. Akad. Amst.

**35**(1932), 525-542.

W. Pauli, “On the formulation of the laws of Nature with five homogeneous components. Part I: classical theory,” Ann. Phys. (Leipzig)

**18**(1933), 305-336; “Part II: the Dirac equation for matter waves,” 337-372.

O. Veblen, “Projective relativity theory,”

*Ergebnisse der Mathematik*, v. 2, Springer, Berlin, 1933.

J. A. Schouten, "Projective relativity," Ann. de l'Inst. Henri Poincare

**5**(1935), 51-88.

E. Cartan, "The method of moving frames, the theory of continuous groups, and generalized spaces," Exposes de Geometrie, t. V, Hermann, Paris, 1935.

G. Ludwig,

*Progress in the Projective Theory of Relativity*, Friedr. Wieweg and Son, Braunschweig, 1951.

O. Pylarinos, "On the differential geometry of ruled surfaces," Annali di Mat. pura ed appl. (4)

**36**(1970), 389-412.

W. Wunderlich, "Ruled surfaces of fixed twist with constantly-twisted bands of striction," Czech. Math. J.

**31**(1981), 457-468.

__Anholonomic geometry__

Abbot Issaly, "Geometric study of the curvature of pseudo-surfaces," Bull. Soc. Math. France

**17**(1889), 84-101.

R. von Lilienthal, "On the shortest integral curves of a Pfaff equation," Math. Ann.

**52**(1899), 417-432.

A. Sommerfeld, review of the book

*Grundlagen einer Krummungslehre der Curvenscharen,*by R. von Lilienthal (Teubner, Leipzig, 1896), published in Gottingsche gelehrte Anzeigen

**160**(1898), 901-912.

D. Sintzov, "On the curvature of the integral curves of the Pfaff equation," Math. Ann.

**101**(1928), 261-272.

A. Wundheiler, "Rheonomic geometry. Absolute mechanics," Prac. Matemalyczno-Fizycznych

**40**(1932), 92-142.

G. Vranceanu, “Anholonomic spaces,”

*Mémoires des sciences mathématiques*, LXXVI, Gauthier-Villars, Paris 1936.

__Other topics__

C. Cattaneo, “Natural projection and transverse derivation in a Riemannian manifold with normal hyperbolic metric,” Ann. di Mat. Pura e Appl.

**48**(4) (1959), 361-386.

C. Ehresmann, “The prolongations of a differentiable manifold. I. Calculus of jets, principal prolongation,” Comptes rendus

**233**(1951), 598-600; “II. The space of jets of order

*r*of

*Vn*into

*Vm*,” pp. 777-779; “III. Transitivity of prolongations,” pp. 1081-1083; “IV. Contact elements and enveloping elements,”

**234**(1952), 1028-1030.