Examples of application of transversality of submanifolds

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Thread starter aboutammam
Start date Jun 30, 2017
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In summary, the concept of transversality of submanifolds is a fundamental principle in differential geometry that describes the intersection of two submanifolds in a higher-dimensional space. It has many applications in various fields, such as physics, engineering, and computer science. It can be used to solve practical problems, but there are limitations to its application. It is closely related to other concepts in differential geometry and plays a crucial role in the study of topology and differential topology.

Jun 30, 2017

aboutammam

Hi,
I am a PhD student, I work on transversal submanifolds. I want to know if someone has an example of application of transversality in physics or ingenning or in any other area, I hear that there is an application in construction of plans but I didn't find it.
Thank you in advance.

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Jul 1, 2017

aboutammam

no answer!

1. What is the concept of transversality of submanifolds?

The concept of transversality of submanifolds is a fundamental principle in differential geometry that describes the intersection of two submanifolds in a higher-dimensional space. It states that if two submanifolds are transverse, their intersection will be a lower-dimensional submanifold.

2. How is transversality of submanifolds applied in real-life situations?

Transversality of submanifolds has many applications in various fields, such as physics, engineering, and computer science. Some examples include analyzing the intersection of surfaces in 3D space, determining the stability of dynamical systems, and studying the behavior of solutions to partial differential equations.

3. Can transversality of submanifolds be used to solve practical problems?

Yes, transversality of submanifolds can be used to solve practical problems in various fields. For example, it can be applied in robotics to plan efficient paths for robots, in computer graphics to create realistic 3D models, and in economics to analyze market equilibria.

4. Are there any limitations to the application of transversality of submanifolds?

Like any mathematical concept, there are limitations to the application of transversality of submanifolds. It may not always be possible to determine the intersection of two submanifolds, or the intersection may not be a submanifold at all. Additionally, the concept is limited to smooth submanifolds and may not apply to more general objects.

5. How does transversality of submanifolds relate to other concepts in differential geometry?

Transversality of submanifolds is closely related to other concepts in differential geometry, such as tangent spaces, vector fields, and differential forms. It also plays a crucial role in the study of topology and differential topology, where it is used to define important notions such as intersection numbers and degree of maps.