# Symmetry (killing vector) preserving diffeomorphisms

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In summary, symmetry preserving diffeomorphisms are transformations that preserve the symmetry of a given system. They play a crucial role in physics, particularly in the study of spacetime, and are used to describe the symmetries present in physical systems. A killing vector is closely related to symmetry preserving diffeomorphisms as it is used to generate infinitesimal transformations that preserve the system's symmetry. Some examples of symmetry preserving diffeomorphisms include rotations, translations, and reflections in Euclidean space. In the study of spacetime, these transformations allow for a deeper understanding of the fundamental laws of physics by identifying and describing the symmetries present.
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Suppose that on a Riemannian manifold (M,g) there is a killing vector such that
##\mathcal{L}_{\xi} g = 0.##

How would one then characterize the group of diffeomorphisms ##f: M \to M## such that

$$\mathcal{L}_{f^* \xi} (f^*g) = 0?$$

How would one describe them? Do they have a name and can an explicit form be found?

Alternatively, given a killing vector ##\xi## how would one describe the diffeomorphisms ##f: M \to M## such that ##\xi## remains a killing vector also for ##f^* g##? I.e. ##\mathcal{L}_{\xi} f^*g = 0##.

## What is the definition of symmetry preserving diffeomorphisms?

Symmetry preserving diffeomorphisms are transformations that preserve the symmetry of a given system. In other words, they do not change the overall structure or behavior of the system.

## How do symmetry preserving diffeomorphisms relate to physics?

Symmetry preserving diffeomorphisms play a crucial role in physics, particularly in the study of spacetime. They are used to describe the symmetries that are present in physical systems and help to simplify and understand complex physical phenomena.

## What is a killing vector and how does it relate to symmetry preserving diffeomorphisms?

A killing vector is a vector field that preserves the metric of a given space. In other words, it is a vector that does not change the distance between points in the space. Killing vectors are closely related to symmetry preserving diffeomorphisms as they are used to generate infinitesimal transformations that preserve the symmetry of a given system.

## What are some examples of symmetry preserving diffeomorphisms?

Some examples of symmetry preserving diffeomorphisms include rotations, translations, and reflections in Euclidean space. In physics, symmetries such as time translation, spatial translation, and rotations are also important symmetry preserving diffeomorphisms.

## Why are symmetry preserving diffeomorphisms important in the study of spacetime?

Symmetry preserving diffeomorphisms allow us to understand the underlying structure and behavior of spacetime. They help us to identify and describe the symmetries that are present, which can ultimately lead to a deeper understanding of the fundamental laws of physics.

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