Definitions
from The American Heritage® Dictionary of the English Language, 5th Edition.
 noun Equality of measure.
 noun Equality of elevation above sea level.
 noun Mathematics A function between metric spaces which preserves distances, such as a rotation or translation in a plane.
 noun Biology A proportional change in the size of a part or parts of an organism as the organism grows.
from Wiktionary, Creative Commons Attribution/ShareAlike License.
 noun mathematics A
function betweenmetric spaces (or on a single metric space) having the property that thedistance between two images is equal to the distance between their preimages.
from WordNet 3.0 Copyright 2006 by Princeton University. All rights reserved.
 noun equality of elevation above sea level
 noun equality of measure (e.g., equality of height above sea level or equality of loudness etc.)
 noun the growth rates in different parts of a growing organism are the same
 noun a onetoone mapping of one metric space into another metric space that preserves the distances between each pair of points
Etymologies
from Wiktionary, Creative Commons Attribution/ShareAlike License
Examples

The metric has time dependent metric elements, which means there is no Killing isometry which defines energy conservation as a symmetry of the spacetime.

Temperaturemediated transitions between isometry and allometry in a colonial, modular invertebarte.

Further momentum is an isometry of this spacetime, so momentum is conserved.

The metric coefficients are time dependent, and this precludes the existence of a Killing vector K_t that defines an isometry and energy conservation law.

In effect cosmologies in their lack of such isometry permits the creation of momentum and energy — aka the big bang.
Everything You Ever Wanted to Know About Quantum Mechanics, But Were Afraid to Ask

With a few [extremely nongeneric] exceptions, namely the ones with maximal isometry groups, one can say definitely whether they are expanding or not.

This process of being dragged along is due to a symmetry (isometry) of space which preserves momentum.

(M², g²ab), respectively, and if there exists an isometry

Does it follow, as we would want determinism to guarantee, that Î¨ is extendible to an isometry from

(M*, g*ab) violates the condition that determinism was supposed to guarantee as Î¨ is not extendible to an isometry from D+ (Î£) onto
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