DIFFERENTIAL GEOMETRY embodies the theorems concerning curves, surfaces and other manifolds which involve applications of the calculus. Straight lines, circles, planes and spheres are geometrical entities possessing the common property that any part of one of them has the same shape as any other part. Other curves and surfaces do not possess this property, as for example, conics, ellipsoids and paraboloids. The geometrical character of these entities varies in general continuously from point to point, and consequently the calculus is needed in order to study many of their geometrical properties.
Most of the older differential geometry, and much of the recent, in so far as magnitudes are concerned, rests upon the assumption that the curves and surfaces under consideration lie in ordinary, or Euclidean, space of three dimensions, and that the measurement of magnitudes, such as lengths of curves, angles and areas of sur faces, is based upon Euclidean measure. Thus, when the three dimensional space is referred to rectangular cartesian coordinates, the quadratic differential form ds2 = (dx)2+ (dy)2+ (d2)2, (1) which defines the square of the distance between points (x, y, z) and (x+dx, y+dy, z+dz), leads by the processes of the inte gral calculus to the determination of the length of a curve between two points of it. In fact, a curve is defined by two equations of the form f(x, y, z) =o, g(x, y, z) =o. (2) For differentials dx, dy, dz in the direction of this curve and by means of (2), equation (I) is reducible to the form ds=F(x)dx, and then the length of arc is given by an integration. Or any curve may be defined, in many ways, in the form x =b(t), Y = f2(l) , z = h(t) where t is a parameter, and then the length of arc is given by an integral in t.
When all the points of a curve lie in one plane, it is called a plane curve, otherwise a skew curve. If P is any point on a skew curve C, l the tangent line to C at P, and Q is any other point of the curve, the plane determined by l and Q assumes a limiting position as Q approaches P along the curve ; it is called the oscu lating plane to C at P; of all the planes through l it lies nearest to the curve in the sense that the distance of a point of C near P from the osculating plane is of the second order and from the other planes it is of the first order. The normal to C at P in the osculating plane is called the principal normal, and the nor mal to the osculating plane at P the binormal. The manner in which the configuration consisting of the tangent, principal normal, and binormal varies in direction, as the point describes the curve, characterizes the curve. The rate of change of direction of the tangent with the arc is the curvature, and of the binormal the torsion. These are the fundamental elements in the differential geometry of curves and in any of the many treatises the reader will find extensive developments of the theory.
A surface is a locus of two dimensions. A surface in Euclidean space, or a portion of it, is defined by one relation between the coordinates as f(x, y, z) =o. (3) If P is an ordinary point of the surface, not a singular point such as the vertex of a cone, the tangents at P to all curves on the sur face through P lie in a plane, called the tangent plane at P; its equation is (X—x)+ of (Z—z) =o. (4) ax az When one of the three variables x, y, z is eliminated from (4) by means of (3), ordinarily the other two enter in the equation, that is, the tangent plane depends upon two parameters. When the tangent plane involves a single parameter, the surface is called developable, otherwise non-developable; a developable surface can be rolled out, or developed, upon a plane.
When the tangent plane at an ordinary point P of a non-develop able surface is taken for the plane z=o, and any two orthogonal lines in this plane are taken for the x and y axes, the equation of the surface, at least in the neighbourhood of the point, can be written in the form y), where c (x, y) is a power-series in x and y of the third and higher orders, and a, b and c are constants. The ellipse or hyperbola whose equations are axe+ = I, z=o I, z=o, is called the Dupin indicatrix of the surface at P. The principal axes of the conic are called the principal directions at P; conju gate diameters determine conjugate directions, and the asymptotes asymptotic directions. Two one-parameter families of curves on the surface whose directions at a point of meeting are conjugate are said to form a conjugate system; a curve whose direction at every point is asymptotic is any asymptotic line; a curve whose direction at every point is principal is a line of curvature. These various curves may be defined also by properties involving the tan gent plane. The tangent planes to a surface along any curve form a developable surface, and the directions of the generators are conjugate to the given curve; the tangent planes to a surface along an asymptotic line are the osculating planes of the latter ; the normals to a surface along a line of curvature are tangent to a curve in space, and this is true only of a line of curvature.
If in equation (3) we put x and y each equal to a function of two parameters, a and v, and solve for z, the surface is defined by three equations x =f1(u, v), y=f2(u, v), z =f3(u, v) • (5) Conversely, any three equations of this type define a surface. This method of definition is due to Gauss. Owing to the great arbitrari ness in the choice of the parameters u and v, it is a very powerful method and has simplified the solution of many problems. When the expressions (5) are substituted in (I) , we get the differential form where E, F and G involve the first derivatives of f 1j f 2, f Any curve on the surface is defined by a relation between u and v ; when this is used in connection with (6), the latter defines the length of the curve. The right-hand member of (6) is called the first fundamental quadratic form of the surface. There is another quadratic differential form of importance, called the second funda mental form and usually written (7) to within terms of third and higher orders it is equal to twice the distance from the point (u+du, v+dv) on the surface to the tangent plane at the point (u, v). Measurement of angles be tween directions at a point on the surface depends only upon the first form, and conjugacy only upon the second. Since lines of curvature form an orthogonal system and also a conjugate sys tem, their differential equation involves the coefficients of both forms.
Through each point of a surface, and in each direction, there is a curve of the surface whose principal normal at each point is normal to the surface. Although this definition involves a property of the curve as viewed from the Euclidean space in which the sur face is contained, the differential equations of these curves involve only the coefficients of the first form and their first derivatives, that is, they are characterized by a property of distance alone. In fact, they are the curves for which the first variation of the inte gral j ds is zero; that is, they are the extremals of this integral, to use the terminology of the calculus of variations. These curves are called the geodesics of the surface.
The foregoing are the fundamental elements which enter into the vast body of theorems concerning surfaces in a Euclidean space, which are to be found in any of the many treatises. Much of the theory involves metric properties. Since conjugate systems and asymptotic lines are invariant under projective transforma tions of the enveloping Euclidean space, there is a considerable theory which is projective in this sense. It is a geometrical inter pretation of linear partial differential equations of the second order. This has been developed by Darboux, Guichard, Tzitzeica, Demoulin and Wilczynski. Recently Fubini and Bompiani have made further developments.
BIBLIOGRAPHY.-W. Blaschke, Vorlesungen fiber Di ff erentialgeoBibliography.-W. Blaschke, Vorlesungen fiber Di ff erentialgeo- metrie (Berlin, 1921) ; L. Bianchi, Lezioni di geometria differenziale, 2 vols. (Pisa, 1922-23) ; G. Darboux, Lecons sur la theorie generale des surfaces, 4 vols. (Paris, 1887-96 ; 2nd ed., 2 vols., 1914-15) ; L. P. Eisenhart, Differential Geometry (Boston, 1909) and Trans formations of Surfaces (Princeton, 1923) ; A. R. Forsyth, Differential Geometry (Cambridge, 1912) ; G. Fubini and tech, Geometria proiettiva differenziale, 2 vols. (Bologna, 1926-27) ; G. Scheffers, Anwendung der Differential- and Integral-Rechnung auf Geometrie, 2 vols. (Leipzig, 1901-02) ; G. Tzitzeica, Geometrie differentielle pro jective des reseaux (Bucharest, 1924) ; C. E. Weatherburn, Differential Geometry of Three Dimensions. (Cambridge, 1927) ; E. J. Wilczynski, Projective Differential Geometry of Curves and Ruled Surfaces (Leipzig, 1go6) . (L. P. E.)