# Differential Geometry

• I. N. Bronshtein
• K. A. Semendyayev

## Abstract

In differential geometry we investigate plane or space curves and surfaces by using the methods of differential calculus. Therefore we assume that the functions involved in the equations are continuous and have continuous derivatives up to a certain order which is needed in the considered problem(1). In dealing with geometrical objects given by their equations, we distinguish those properties which depend on the choice of a coordinate system (as, for example, points of intersection of the curve with the coordinate axes, the slope of a tangent line, maxima and minima) and invariant properties which are not disturbed by transformations of coordinates and which therefore depend only on the curve or surface itself (as, for example, points of inflection, vertices or curvature of a curve). On the other hand, we distinguish the local properties which concern only small parts of a curve or a surface (e.g., curvature, linear element of a surface) and the properties of a curve or surface in the whole (e.g., number of vertices, length of a closed curve).

## Keywords

Singular Point Positive Direction Fundamental Form Tangent Plane Tangent Line
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## Notes

1. (1).
This condition may fail only at certain separate points of a curve or surface; in this case we have points of a special type (as, for example, a discontinuity or a bend of a curve). For such points see pp. 285, 306.Google Scholar
2. (2).
For the general notion of an equation of a line see p. 239.Google Scholar
3. (1).
For the differential, see pp. 362–364.Google Scholar
4. (1).
More precisely, it is convex in the positive direction of the y axis.Google Scholar
5. (1).
We shall discuss here only those points which are invariant with respect to transformations of coordinates. Maxima and minima are discussed on pp. 379–383.Google Scholar
6. (1).
For determining the points of inflection in which f′(x) does not exist (for example, becomes infinite) see below.Google Scholar
7. (1).
8. (1).
The degree of a term Axmyn is the sum m + n of the exponents. E.g., the term 3x2y2 is of degree 5, the term 2y2 is of degree 2; the highest terms of the polynomial x3 + y3-3xy are x3 and y3 (both of degree 3).Google Scholar
9. (1).
The helix defined by these equations and shown in Fig. 250 is called right handed. An observer looking at the helix down its axis (the z axis) sees the line ascending counterclockwise. A helix symmetric to the right handed helix with respect to a plane is left handed.Google Scholar
10. (1).
For notation p, q, r, s, t, see p. 309.Google Scholar
11. (1).
For notation p, q, r, s, t, see p. 309.Google Scholar
12. (1).
For notation p, q, r, s, t, see p. 303.Google Scholar