Preservation of the angle between two vectors

[LCSphysicist]

Homework Statement:: .
Relevant Equations:: .

Generally, when we talk about preservation of angle between two vectors, we talk about conformal transformation. But what is confusing me is, shouldn't any general transformation of coordinates preserve the angle between two vectors?

What i mean is: The expression for the angle is given by $$cos(\theta) = \frac{ V^{\mu}U^{v} g_{\mu v}}{\sqrt{(V^{a}V^{b} g_{ab})(U^{r}U^{s} g_{rs})}}$$

Isn't it automatically invariant? So why do we bother to study in detail (even given them a name, conformal transformation), if all transformation preserves it after all?

 

[Orodruin]

That's for general coordinate transformations. What is considered here is the pushforward of the vectors without using the pullback of the metric defined by the inverse of the conformal transformation.

 

[FactChecker]

Since this is posted in General Math, it should be said that a general transformation does not usually preserve angles. The transformation, $x \rightarrow x; y \rightarrow 2y$ does not preserve angles. There may be specific contexts where the angel is preserved, but not in general mathematics.

 

[Orodruin]

Since this is posted in General Math, it should be said that a general transformation does not usually preserve angles. The transformation, $x \rightarrow x; y \rightarrow 2y$ does not preserve angles. There may be specific contexts where the angel is preserved, but not in general mathematics.

Well … it does preserve angles if you use the metric induced by the pullback of the inverse of the transformation. It does not preserve angles if you use the metric already present.

When discussing what ”preserves angles” mean, it is important to consider what metric is being used in each case to make the comparison.

 

[FactChecker]

Well … it does preserve angles if you use the metric induced by the pullback of the inverse of the transformation. It does not preserve angles if you use the metric already present.

When discussing what ”preserves angles” mean, it is important to consider what metric is being used in each case to make the comparison.

That is an interesting comment. I am used to the complex analysis context, where the metrics are established and standard. There, most general transformations are not conformal.

 

[Orodruin]

That is an interesting comment. I am used to the complex analysis context, where the metrics are established and standard. There, most general transformations are not conformal.

Most transformations are not conformal unless you bring the metric with you in the sense of also transforming the metric.

 

[Svein]

Unstated assumptions:

• You are talking about metric spaces
• The expression given in the OP presumposes a Hilbert space (yes, Hilbert spaces are metric, but metric spaces do not necessarily contain an inner product)

 

[Office_Shredder]

@Svein How do you measure an angle in a space that doesn't have an inner product defined?

 

[Svein]

@Svein How do you measure an angle in a space that doesn't have an inner product defined?

Yes, that was what I was wondering about too. I suspect the OP was thinking about ℝⁿ for some n (n=3?).

 

[WWGD]

Iirc, there is a generalized notion of angle for Metric Spaces . I think it was in BBI *'s " Metric Geometry"

* Burago, Burago , Ivanov.