- #1
precondition
- 57
- 0
Could someone explain these two concepts? What I need is the big picture of 'why' we need this, roughly 'what' these equations mean, 'where' its used etc
Thanks in advance
Thanks in advance
In summary, The concepts being discussed are Cartan forms and connections, which are used to define differentiation on certain geometrical objects called fibre bundles. These concepts have proven useful in constructing models for phenomena in General Relativity and string theory. More information can be found in books on gauge theory and physics, or a mathematical treatment can be found in "Cartan for Beginners: Differential Geometry Via Moving Frames and Exterior Differential Systems" by Tom Ivey.
- #2
precondition
- 57
- 0
omg.. nobody answering...
there aren't many people who understand these stuff eh?
there aren't many people who understand these stuff eh?
- #3
Doodle Bob
- 255
- 0
precondition said:omg.. nobody answering...
there aren't many people who understand these stuff eh?
i think it is more of a matter of complexity. there are entire books written to answer that question.
I'll give the why, you can explore the what on your own: Cartan forms and connections allow us to define differentiation on certain geometrical objects, namely fibre bundles. This topic has proven valuable in constructing models for phenomena in GR and string theory.
Any book on gauge theory and physics will discuss this. for a heavy duty mathematical treatment, look for the book Cartan for Beginners: Differential Geometry Via Moving Frames and Exterior Differential Systems by my friend Tom Ivey.
Related to Cartan forms and structure equations
What are Cartan forms and structure equations?
Cartan forms and structure equations are mathematical tools used in differential geometry to describe the curvature and geometry of a manifold. Cartan forms are differential forms that encode the geometric properties of the manifold, while structure equations are equations that relate the Cartan forms to the curvature and torsion of the manifold.
What is the purpose of Cartan forms and structure equations?
The purpose of Cartan forms and structure equations is to provide a systematic and elegant way of studying the geometry of a manifold. They allow us to calculate the curvature and torsion of a manifold, which are important quantities in understanding its intrinsic properties. They also provide a powerful framework for studying gauge theories and other physical theories that involve differential geometry.
How are Cartan forms and structure equations related to Lie groups?
Cartan forms and structure equations are closely related to Lie groups, which are groups of symmetries that preserve the geometric properties of a manifold. In fact, Cartan forms can be thought of as the "infinitesimal generators" of a Lie group, while structure equations describe how these generators combine to give rise to the curvature and torsion of the manifold.
What are some applications of Cartan forms and structure equations?
Cartan forms and structure equations have a wide range of applications in mathematics and physics. In mathematics, they are used in the study of differential geometry, Lie groups, and algebraic topology. In physics, they have applications in general relativity, gauge theories, and string theory. They also have practical applications in engineering, such as in the design of control systems and robotics.
Are there any limitations to using Cartan forms and structure equations?
Like any mathematical tool, there are limitations to using Cartan forms and structure equations. They are most useful for studying smooth manifolds, and may not be applicable to rough or discontinuous surfaces. They also require a certain level of mathematical background and expertise to understand and use effectively. Additionally, some problems in differential geometry may be better approached using other techniques, depending on the specific problem at hand.
Similar threads
-
Differential Geometry
-
Differential Geometry
-
Differential Geometry
-
Differential Geometry
-
Differential Geometry
-
Special and General Relativity
-
Mechanical Engineering
-
Differential Geometry
-
Biology and Medical
-
Differential Geometry