Projective Differential Geometry Research Articles

A note on the projective differential geometry of a non-holonome surface

The paper is concerned with projective differential properties of the plane sections made by the planes which pass through a non-asymptotic tangent, of a non-holonome surfaceV32 inS3. Chiefly by this way some surfaces are determined, which are projectively connected with the generic point on theV32.

On the projective differential geometry of a non-holonome surface in ordinary space

The paper deals with many projective properties of the neighbourhoods of the third,4th, 5th order of a point on a non-holonome surfaceV32 inS3. Chiefly two remarkable projective correspondences are studied, between the tangent plane and the bundle of directions, with his contact point as centre. Many generalizations are obtained of geometrical loci (so as lines, planes, quadrics) projectively associated with the neighbourhoods of a point on an ordinary (holonome) surface. The last sections are concerned with the extension to theV32 of the quadric ofMoutard and theSegre's correspondence.

Abstract flat projective differential geometry

Abstract flat projective differential geometry

Note on the projective differential geometry of space curves

Note on the projective differential geometry of space curves

General Projective Differential Geometry

General Projective Differential Geometry

On a new Definition of Vector and Parallel Displacement in Projective Differential Geometry

On a new Definition of Vector and Parallel Displacement in Projective Differential Geometry

Projective differential geometry of curves

Projective differential geometry of curves

Consecutive covariant configurations at a point of a space curve

A systematic study of the projective differential geometry of space curves was first made by Halphent in a memoir of 1880. Wilczynskil in 1905 and 1906 and Sannia? in 1926 made important additions to the subject. The projective differential theory of a curve involves many configurations associated covariantly with the curve. The purpose of this paper is to make some contributions to the theory of a space curve, which are based upon the study of configurations. The work follows the lines of a similar investigation made by Lane I for the case of a plane curve. We now make precise the meaning of the word consecutive as used in the present paper. Let us consider an analytic curve C in projective space of three dimensions. The equations of such a curve, in non-homogeneous projective coordinates x, y, z, can be written in the form of two power series expansions,

Projective Differential Geometry

(1933). Projective Differential Geometry. The American Mathematical Monthly: Vol. 40, No. 10P1, pp. 568-579.

Projective Differential Geometry

Projective Differential Geometry

BOOK REVIEWS

Book reviewed in this article:A Course in General Chemistry, including an Introduction to Qualitative Analysis, for use in colleges, by William C. Bray, Professor of Chemistry in the University of California, and Wendell M. Latimer, Professor of Chemistry in the University of California.Principles of General Chemistry, by Stuart R. Brinkley, Associate Professor of Chemistry, Yale UniversityVolumetric Analysis, by G. Fowles, M.Sc., A.I.C., Assistant Master, Latymer Upper School, Mammersmith.Plane Trigonometry, by Cecil A. Ewing, Head of the Mathematics Department in the Tome School, Port Deposit, Maryland.Intelligence: Its Manifestations and Measurement, by Paul L. Boynton, George Peabody College for TeachersOther Worlds, by Edwin Lincoln Moseley, Head of the Department of Biology, Ohio State Normal College.A Textbook of Physics, Vol. I. Mechanics by E. Grimsehl, edited by R. Tomaschek, University of Marburg.Projective Differential Geometry of Curves and Surfaces, by Ernest Preston Lane, Professor of Mathematics in the University of Chicago.The Development of Physical Thought, by Leonard B. Loeb, Professor of Physics, University of California, and Arthur S. Adams, Professor of Mechanics, Colorado School of Mines.Science Problems of Modern Life, Book II, by Ellsworth S. Obourn, Head of the Science Department, John Borroughs School, Clayton, Missouri, and Ellwood D. Heiss, Head of the Science Department, State Teachers College, East Stroudsburgh, Pennsylvania.Introductory Mathematics, by John Wayne Lasley and Edward Tankard Browne, Professors of Mathematics, University of North Carolina.Mathematical Excursions, by Helen Abbot Merrill, Professor of Mathematics, Emeritus, in Wellesley College, Wellesley, Mass.Number the Language of Science, by Tobias Dantzig, Ph.D., Professor of Mathematics, University of Maryland; Lecturer on Mathematical Physics, U. S. Bureau of Standards.

Projective Differential Geometry of Curves and Surfaces. By E.P. Lane. Pp. xi, 321. 22s. 1932. (University of Chicago Press)

Projective Differential Geometry of Curves and Surfaces. By E.P. Lane. Pp. xi, 321. 22s. 1932. (University of Chicago Press)

Book Review: Projective Differential Geometry of Curves and Surfaces

Book Review: Projective Differential Geometry of Curves and Surfaces

A complete system of tensors of linear homogeneous second-order differential equations

In an article appearing in the American Mathematical Monthlyt Lane called attention to the need in projective differential geometry of an absolute differential calculus for forms of higher degree than the quadratic. Such a calculus for forms of arbitrary degree had been published by the writer4 and it seemed that some adaptation to the theory of projective differential geometry would be desirable. However, it appeared more satisfactory to base a tensor theory directly upon the differential equations. Inasmuch as various systems of differential equations are used in the projective theory, a separate tensor theory will need to be constructed for each type. In this paper a study has been made of the differential equations upon which the ruled surface theory is based. The remarkable simplicity which has been introduced by tensor methods in the theory of Riemannian geometry and of algebraic invariants appears also in this theory. The Wilczynski theory, while general, has had the blemish of containing awkward unsymmetrical sets of equations involving cumbersome numerical constants. And with the improvement in notation there is an accompanying advantage in method, the simple formal tensor processes replacing the earlier ingenious methods. Some geometric interpretations of the tensors and invariants are given here in which the dependent variables yi(i = 1, * * * , n) are interpreted as nonhomogeneous co6rdinates of a point. This is in contrast to the convention peculiar to projective differential geometry of choosing a fundamental set of 2n independent solutions (y', * * , y), o =1, * * , 2n, and interpreting these as n points (y * * yn)), i 1, .. , n, in a homogeneous space of 2n-1 dimensions. An interpretation of our tensors in this latter space will constitute a generalization of Wilczynski's ruled surface theory.

Book Review: Bibliography of Projective Differential Geometry

Book Review: Bibliography of Projective Differential Geometry

Introduction à la géométrie projective différentielle des surfaces

ALTHOUGH the metrical part of differential geometry has been studied for two hundred years, the projective aspect received little attention until much later. Perhaps the earliest work on the subject was that of G. H. Halphen in 1875, but the systematic investigation of the projective differential geometry of surfaces was commenced by Wilczynski about 1906. Since then a great deal has been done, especially in Italy and the United States. Introduction à la géométrie projective différentielle des surfaces. Par Prof. Guido Fubini Prof. Eduard Čech. Pp. vi + 291. (Paris: Gauthier-Villars et Cie, 1931.) 60 francs.

On the Projective Differential Geometry of Conjugate Nets

On the Projective Differential Geometry of Conjugate Nets

A Note on the Projective Differential Geometry of a Surface

A Note on the Projective Differential Geometry of a Surface

A canonical form of Green’s projective analogue of the Gauss differential equations

The projective differential geometry of a surface in space of three dimensions may be founded upon Green's projective analogue of the Gauss differential equations.t In this paper we assume for the parametric curves any non-asymptotic system, and derive a canonical form for Green's equations. The method used is an adaptation of the method used by Lane in deriving a canonical form: for the defining differential equations of a conjugate net. By specializing the parametric curves to be conjugate, we may obtain the equivalent of Lane's canonical form. Again by assuming the parametric curves to be non-conjugate we obtain a canonical form for Green's differential equations? defining a non-conjugate net. These canonical forms may be written with covariant derivatives.

Tetrads of ruled surfaces

The projective differential geometry of a configuration composed of three ruled surfaces whose generators are in correspondence in sets of three has been discussed in two recent papers by the author. t It is the purpose of the present paper to present briefly an analytic basis for the projective differential geometry of a configuration composed of four ruled surfaces whose generators correspond in sets of four, each set containing one line element from each surface. This correspondence is brought about by choice of a parameter common to all four surfaces. For defining system we choose a set of eight ordinary first-order linear and homogeneous differential equations in eight dependent variables, together with four linear and homogeneous equations of order zero: