1. If F: M → N is a mapping, show that the following conditions on its tangent map at one point p are logically equivale

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1 If F M N Is A Mapping Show That The Following Conditions On Its Tangent Map At One Point P Are Logically Equivale 1 (56.53 KiB) Viewed 28 times

1. If F: M → N is a mapping, show that the following conditions on its tangent map at one point p are logically equivalent: (a) F* preserves inner products. (b) F* preserves lengths of tangent vectors, that is, || F*(v) || = || v || for all v at p. (C) F* preserves frames: If e, e, is a tangent frame at p, then F* (e), F* (e) is a tangent frame at F(p). а 6.4 Isometries and Local Isometries 287 (d) For some one pair of linearly independent tangent vectors v and wat P, . ||F*(v)|| = |v||||F*(w)|| = ||w|| and F*(v) • F#(w) = v • w. [Hint: It suffices, for example, to prove (a) = (c) = (d) = (b) = (a).] These are general facts from linear algebra; in this context they provide useful criteria for F to be a local isometry.