By Sebastian Montiel and Antonio Ros

ISBN-10: 0821847635

ISBN-13: 9780821847633

This introductory textbook places forth a transparent and concentrated perspective at the differential geometry of curves and surfaces. Following the fashionable viewpoint on differential geometry, the booklet emphasizes the worldwide points of the topic. the wonderful choice of examples and routines (with tricks) may help scholars in studying the cloth. complex undergraduates and graduate scholars will locate this a pleasant access aspect to differential geometry.

In order to review the worldwide houses of curves and surfaces, it is vital to have extra subtle instruments than are typically present in textbooks at the subject. particularly, scholars should have a company grab on yes topological theories. certainly, this monograph treats the Gauss-Bonnet theorem and discusses the Euler attribute. The authors additionally conceal Alexandrov's theorem on embedded compact surfaces in R3 with consistent suggest curvature. The final bankruptcy addresses the worldwide geometry of curves, together with periodic area curves and the four-vertices theorem for airplane curves that aren't unavoidably convex.

Besides being an creation to the full of life topic of curves and surfaces, this booklet can be used as an access to a much wider examine of differential geometry. it's appropriate because the textual content for a first-year graduate direction or a complicated undergraduate course.

Readership: Undergraduate scholars, graduate scholars, and study mathematicians attracted to the geometry of curves and surfaces.

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**Additional info for Curves and Surfaces**

**Example text**

Given p E S and v E TPS, for each curve a : (-E, s) -* S such that a(0) = p and a'(0) = v, we have (df)(v) _ (f o a)'(0) _ (F o a)'(0) _ (dF)(v). 59. Let S be a surface, p E S, v E TPS, and let a : (-E, s) -* S be a curve such that a(0) = p and a'(0) = v. A: (Height function) If po E I[83 and a E I[83 - {0}, then the differential of the height function h : S - Il8 given by h(p) _ (p - po, a) is (dh)(v) - dt It_o (a(t) - poi a) _ (a'(O),a) _ (v,a). 60. We compute here the differential for inclusions, constant maps, and restrictions.

L. is an arc of a circle if and only if all its normal lines pass through a given point. l. is a segment of a straight line or an arc of a circle if and only if all its tangent lines are equidistant from a given point. , prove that all the normal lines of a are equidistant from a point if and only if there are a, b E l[8 such that 1 as + b for each s E I. l. and let so E I. We define the function f : I -* l[8 by f(s) = (cr(s) - c(so), N(so)), measuring the oriented distance from the point a(s) to the tangent line of a at so.

L. and let M : a rigid motion. If we set /3 = M o cx, show that -* R3 be T«(s) for each s E I, if M is direct, -Ta(s) for each s E I, if M is kg(s) - ka(s) and 'r (s) - inverse. 34. l. with positive curvature defined on an interval I C Il8 symmetric relative to the origin. We define another curve /3: I -* Il83 by ,Q(s) = a(-s), for each s E I. l. and that kg(s) = k«(-s) and 'r(s) _ -T«(-s) for each s E I. l. l. with positive curvature from its curvature and its torsion. 21 of the local theory of plane curves was a constructive proof.

### Curves and Surfaces by Sebastian Montiel and Antonio Ros

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