By Chuan-Chih Hsiung

The origins of differential geometry return to the early days of the differential calculus, while one of many basic difficulties was once the selection of the tangent to a curve. With the advance of the calculus, extra geometric purposes have been got. The valuable members during this early interval have been Leonhard Euler (1707- 1783), GaspardMonge(1746-1818), Joseph Louis Lagrange (1736-1813), and AugustinCauchy (1789-1857). A decisive breakthrough used to be taken by way of Karl FriedrichGauss (1777-1855) together with his improvement of the intrinsic geometryon a floor. this concept of Gauss was once generalized to n( > 3)-dimensional spaceby Bernhard Riemann (1826- 1866), hence giving upward thrust to the geometry that bears his identify. This e-book is designed to introduce differential geometry to starting graduate scholars in addition to complicated undergraduate scholars (this advent within the latter case is necessary for remedying the weak spot of geometry within the ordinary undergraduate curriculum). within the final couple of a long time differential geometry, besides different branches of arithmetic, has been hugely built. during this e-book we are going to examine purely the normal subject matters, specifically, curves and surfaces in a three-d Euclidean area E3. in contrast to so much classical books at the topic, although, extra consciousness is paid right here to the relationships among neighborhood and international homes, as against neighborhood homes simply. even though we limit our consciousness to curves and surfaces in E3, so much worldwide theorems for curves and surfaces in this booklet will be prolonged to both greater dimensional areas or extra normal curves and surfaces or either. additionally, geometric interpretations are given besides analytic expressions. this can permit scholars to utilize geometric instinct, that's a priceless software for learning geometry and comparable difficulties; this sort of device is seldom encountered in different branches of arithmetic.

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**Extra info for A first course in differential geometry**

**Example text**

2d) coordinates on S2 via stereographic projection onto the planes z = I and z - I , again failing at the south and north pole, respectively, but otherwise being compatible with the previous coordinates. On a manifold we should allow the use of all coordinate systems that are compatible with those that originally were used to define the manifold. Such a collection of compatible coordinate systems is called a maximal atlas. (v) If Mil is a manifold with local coordinates { U ; X I , . . , x " } and is a manifold with local coordinates { V ; Y I , .

2, two surfaces F = ° and G ° in ]R 3 intersect to yield a curve M. The simplest case is one function F of N variables (x I, . . , xN), If at each point of the locus F = c there is always at least one partial derivative that does not vanish, then the Jacobian (row) matrix [aFlax1, aF lax2, " aFlaxN] has rank I and we may conclude that this locus is indeed an (N - I)-dimensional submani fold of ]RN . This criterion is easily verified, for example, in the case of the 2-sphere 2 2 2 F(x, y, z) = x + y + Z - I of Example (ii).

These sets do define a topology for A. For example, let A be a line in the plane � 2 . An open ball in � 2 is simply a disc without its edge. This disc either will not intersect A or will intersect A in an "interval" that does not contain its endpoints. Thi s interval will be an open set in the induced topology on the line A. It can be shown that any open set in A will be a union of such intervals. Any open set in M that contains a point x E M will be called a neighborhood of x . If F : M -+ N is a map of a topological space M into a topological space N , we say that F is continuous if for every open set V e N , the inverse image F- 1 V := (x E M I F (x ) E V } is open in M.