Sep 2, 2019 · If the object X is a set of points in the plane with its metric structure or any other metric space, a symmetry is a bijection of the set to itself which preserves the distance between each pair of points (an isometry). So from this I can conclude that every symmetry is an isometry, but not every isometry is a symmetry.

May 14, 2018 · An isometry is a shape preserving transformation. Rotations and reflections are two examples. A dilation is not an isometry because it changes the size of the shape. An example of an isometry group would be all the transformations of a say a regular hexagon (rotations and reflection) that would result in no change in the appearance (symmetry).

Sep 14, 2017 · The image of a complete metric space under an isometry is itself complete (just think about Cauchy ...

Yeah I thought that too at first, but you can show an isometry of $\mathbb{R}^n$ fixing the origin is linear without assuming that it's surjective. The key is the inner product, which, of course, you don't have in a general normed vector space.

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An isometry, on the other hand, only requires that the columns are orthonormal, but not that they form a basis. It trivially follows that any unitary is also an isometry. In other words, an isometry is a matrix whose columns are orthonormal, while a unitary is a …

Apr 4, 2023 · An isometry preserves distances, so since distinct points are at a positive distance from one another, they are mapped to distinct points in the image. (In particular, an isometry never maps distinct points onto the same point, which is zero distance from itself.)

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Nov 15, 2013 · How do I prove that an isometric embedding is necessarily injective? This is my question. I study topology. Someone help, please; I don't have a clue :(

Jul 25, 2017 · When you showed that there is an isometry from the circle to the uncurved line, you essentially showed that the circle has zero (Riemannian) curvature (as does any 1D Riemannian manifold). However, this "intrinsic" curvature being zero does not mean there is no curvature.