# Is Fibre Bundles Cartan's Generalization of Klein's Erlagen Program?

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In summary, the conversation discusses the generalization of Felix Klein's classification of geometries by Cartan. The concept of fiber bundles with a structure group G is mentioned as a way to formalize this generalization, but it is noted that the main idea is to model spaces locally on Kleinian geometries. The conversation also briefly touches on the motivation for the concept of bundles, such as normal bundles and frame bundles.
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As I understand it, Felix Klein sought to classify geometries with respect to what groups G that respected the structure of the given space X. Lately i read in an article on "the history of connections" by Freeman Kamielle that Cartan wished to generalize this notion. Is it correct to think of Fibre bundles ##(E, \pi, B, F)##= (whole space, projection, base, typical fibre) with a structure group G as the generalization that Cartan came up with? I.e. the whole space E does no longer respect the action of G, but each fibre respects it. Is that the correct understanding?

Not exactly. I tried to read a little bit about this sort of thing a while back, but I'm no expert on it, either. But it's definitely more than just a fiber bundle. There were lots of early examples of fiber bundles, like normal bundles (where, in my opinion, the concept of a bundle finds its best motivation because you are naturally lead there to study the twisting of a neighborhood of the embedding, which you can identify with the normal bundle), and Cartan's moving frames (frame bundles--the canonical example of a principal bundle) that evolved into the bundle concept.

That wasn't Cartan's idea. Fiber bundles are how it's formalized, but the idea was really to model spaces locally on Kleinian geometries. So, it's generalizing differential geometry on the one hand and Kleinian geometries on the other hand.

## 1. What is Cartan's generalization of Klein's Erlagen Program?

Cartan's generalization of Klein's Erlagen Program is a mathematical framework that extends the ideas of symmetry and invariance to more complex structures known as fibre bundles. It allows for a deeper understanding of how symmetries can be applied to different types of spaces, such as curved spaces.

## 2. How does Cartan's generalization differ from Klein's Erlagen Program?

While Klein's Erlagen Program focuses on the symmetries of Euclidean spaces, Cartan's generalization expands this concept to include more abstract spaces and structures. It also introduces the concept of connections, which describe how the local symmetries of a space are related to one another.

## 3. What is the significance of Cartan's generalization?

Cartan's generalization allows for a more comprehensive understanding of symmetries and invariance in mathematics and physics. It has been applied to various fields, including differential geometry, topology, and theoretical physics, and has led to important developments in these areas.

## 4. How does Cartan's generalization relate to fibre bundles?

Fibre bundles are at the core of Cartan's generalization, as they provide a mathematical structure for understanding the relationships between local symmetries of a space. By studying these bundles, we can gain a deeper understanding of the symmetries and invariances present in a given space.

## 5. What are some examples of applications of Cartan's generalization?

Cartan's generalization has been applied to various areas of mathematics and physics, including general relativity, gauge theory, and quantum mechanics. It has also been used to study the symmetries and invariances of curved spaces, such as in the study of cosmology and black holes.

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