Titlebook: Differential Geometry: Manifolds, Curves, and Surfaces; Manifolds, Curves, a Marcel Berger,Bernard Gostiaux Textbook 1988 Springer Science+

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Curves: The Local Theory,In this chapter we study arcs, that is, immersions of open intervals of . into finite-dimensional affine or vector spaces (8.1.1). We define points of an arc and several important objects associated with them: the tangent, the osculating plane and the concavity (section 8.2).

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Plane Curves: The Global Theory,In this chapter we will study immersions of the circle in the plane, called closed curves, and embeddings of the circle in the plane, called simple closed curves. Naturally, the interesting objects are equivalence classes of such immersions or embeddings.

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Differential Equations,mulating these concepts in terms of vector fields and integral curves. In 1.2.6 we prove the local existence and uniqueness of integral curves. We also discuss the problem of extending an integral curve into a maximal one (section 1.3).

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Differentiable Manifolds,changes from one parametrization to another are the essential ingredients in the definition of abstract manifolds (section 2.2), which are the right objects for the study of abstract (and sometimes even concrete) differential geometry.

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A Brief Guide to the Local Theory of Surfaces in , what the dimension of the ambient space, is that the intrinsic geometry of curves is trivial; the metric given by the length of paths on a curve is always the same as the metric on some interval of .. And the “shape”, or “position,” of a curve in . is specified by a mere scalar function, the curvature.

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0072-5285 somewhat enlarged version of the 1972 book Geometrie Differentielle. The second part, chapters 10 and 11, is an attempt to remedy the notorious absence in the original book of any treatment of surfaces in three-space, an omission all the more unforgivable in that surfaces are some of the most common

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