Search results
An isometry from nR. n. to R. is a length-preserving mapping. Defnition 13.1. A function nf : R → R. n is an isometry if |f(u) f(v)| = |u v| for all u, nv ∈ R. Let’s take a look at two key examples. » R. n #» Example 13.2. For a matrix A ∈ O n, the linear transformation → R. n x 7→A# x. is an isometry. » #» t# » R. n ...
isometry is 1 we can divide central isometries into two types. A central isometry is direct if it has determinant 1. It is opposite if it has determinant −1. We can extend these ideas to isometries in general. An isometry is direct if it is a direct central isometry followed by a translation and opposite if it is an opposite central
An isometry on Rn is an invertible map S : Rn ! Rn which preserves distance. The set of all isometries on Rn is denoted by Isom(Rn). 7.2 Theorem: The set of isometries on n. R is a group under composition. Proof: The identity map I : Rn ! Rn is an isometry because I(x) I(y) = kx yk for all x; y 2 Rn.
in the same way to conclude that an isometry in R3 is either a rotation, or a rotation followed by a °ip of the (x;y)-coordinates. Here is an exercise that is surprising easy: Suppose f: R3!R3 is any map that preserves distances. That is, we are not requiring f to be even linear. Show that f = Tv –L where L is a linear isometry,
16.2. RIEMANNIAN COVERING MAPS 751 Because a Riemannian covering map is a local isometry, we have the following useful result. Proposition 16.5. Let ⇡: M ! N be a Riemannian covering. Then, the geodesics of (M,g) are the pro-jections of the geodesics of (N,h) (curves of the form ⇡ , where is a geodesic in N), and the geodesics
15.2. RIEMANNIAN COVERING MAPS 777 In particular, the deck-transformations of a Riemannian covering are isometries. In general, a local isometry is not a Riemannian cover-ing. However, this is the case when the source space is complete. Proposition 15.7. Let ⇡: M ! N be a local isome-try with N connected. If M is a complete manifold,
In words, we say that a function F is an isometry if it preserves distances. Isometries are also called rigid transformations and we view them as the natural family of transformations that preserve the \structure" of Euclidean space. Example: In R2 (as we will prove) every rotation R x; and every mirror M Lis an isometry.
