Titlebook: Elementary Topics in Differential Geometry; J. A. Thorpe Textbook 1979 Springer-Verlag New York Inc. 1979 Differentialgeometrie.Isometrie.

cobble

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MIRE

The Gauss Map,on .:. → ?.. associated with the vector field . by .(.) = (., .(.)), . ∈ ., actually maps . into the unit .-sphere S. ? ?.. since ∥.(.)∥ = 1 for all . ∈ .. Thus, associated to each oriented .-surface . is a smooth map .: . → S.. called the .. . may be thought of as the map which assigns to each poin

ENNUI

Geodesics, proccss of differentiation of vector fields and functions defined along parametrized curves. In order to allow the possibility that such vector fields and functions may take on different values at a point where a parametrized curve crosses itself, it is convenient to regard these fields and functio

Coronation

Parallel Transport,however, generally not tangent to .. We can, nevertheless, obtain a vector field tangent to . by projecting ?(.) orthogonally onto .. for each . ∈ . (see Figure 8.1). This process of differentiating and then projecting onto the tangent space to . defines an operation with the same properties as diff

Gum-Disease

金桌活畫(huà)面

金桌活畫(huà)面

Convex Surfaces,ee Figure 13.1). An oriented .-surface . is . at . ∈ . if there exists an open set . ? ?.. containing . such that . ∩ . is contained either in . or in .. Thus a convex .-surface is necessarily convex at each of its points, but an .-surface convex at each point need not be a convex .-surface (see Fig

languid

Parametrized Surfaces,ure . and (ii) define various integrals over .. We shall now carry out a similar program for .-surfaces (. > 1). It will turn out that oriented .-surfaces (even connected ones) in general admit only local parametrizations, but that will be adequate for our needs.

濕潤(rùn)

陰郁

The Exponential Map, We begin by using a technique of the calculus of variations analogous to the one we used in Chapter 18 to study minimal surfaces. Now, however, we shall vary parametrized curves rather than parametrized surfaces