Books about Kähler manifolds, Ricci flatness and other things

In summary, the speaker is a pensioner with a background in electronics and IT who has always been interested in theoretical physics. They have studied various topics such as quantum physics, general relativity, and group theory, and are now exploring string theory. However, they have come across unfamiliar mathematical concepts such as Kähler manifolds and Ricci flatness and are looking for book recommendations to gain a better understanding. The person they are speaking with suggests Nakahara's book as a good starting point, which makes the speaker feel grateful and happy.
  • #1
StenEdeback
65
38
Summary:: Books about Kähler manifolds, Ricci flatness and similar things

Hi,

I have an MSc degree in electronics, and I worked with IT. I was always interested in theoretical physics and did extra studies of this in my youth. Now as a pensioner I do private studies for fun. I have studied quantum physics, elementary particle physics, general theory of relativity, and group theory, and now I am into string theory. I run into some mathematics that is new to me, like Kähler manifolds, Ricci flatness, Chern classes, canonical bundles, metrics with global holonomy, and similar things. I look at them in Wikipedia, but I feel the need for a more wholesome treatment, that is books about these things. So please give me tips about good books! I will be grateful.
 
  • Like
Likes vanhees71
Physics news on Phys.org
  • #3
Thank you very much! That book looks really good. I will get it and start reading it, hoping to understand it. Now you have made me smile and feel good. Thank you again! :) I attach a sunrise picture with the moon, seen from a window in my home.
 

Attachments

  • 51 IMGP1885.jpg
    51 IMGP1885.jpg
    25.5 KB · Views: 105
  • Like
Likes vanhees71 and Demystifier

1. What are Kähler manifolds?

Kähler manifolds are a type of complex manifold, which is a geometric space that locally resembles Euclidean space. They are equipped with a special structure called a Kähler metric, which combines a Riemannian metric and a complex structure in a compatible way.

2. What is the significance of Ricci flatness in Kähler manifolds?

Ricci flatness is a condition on the curvature of a manifold, specifically the Ricci curvature tensor. In Kähler manifolds, Ricci flatness is equivalent to the vanishing of the first Chern class, which has important implications in algebraic geometry and complex analysis.

3. How are Kähler manifolds related to symplectic manifolds?

Kähler manifolds and symplectic manifolds are closely related, as they both arise from the study of complex geometry. In fact, every Kähler manifold is also a symplectic manifold, but the converse is not always true.

4. What are some applications of Kähler manifolds and Ricci flatness?

Kähler manifolds and Ricci flatness have applications in many areas of mathematics, including algebraic geometry, complex analysis, and differential geometry. They also have connections to physics, particularly in the study of string theory and mirror symmetry.

5. Are there any open problems related to Kähler manifolds and Ricci flatness?

Yes, there are still many open problems in the study of Kähler manifolds and Ricci flatness. Some current areas of research include understanding the behavior of Kähler metrics under degeneration, studying the existence and uniqueness of Ricci flat metrics, and exploring connections to other areas of mathematics.

Similar threads

Geometry I would like suggestions regarding reading about geometry and manifolds
    • Science and Math Textbooks
Replies
16
Views
2K
Calculus Theoretical Multivariable Calculus books
    • Science and Math Textbooks
Replies
8
Views
4K
Books and Plan to Study Relativity (SR and GR) in Detail
    • Science and Math Textbooks
Replies
6
Views
2K
I do not understand stress energy tensor for fluids
    • Advanced Physics Homework Help
Replies
6
Views
970
Ricci form and Kahler manifolds
    • Differential Geometry
Replies
2
Views
5K
Initial Academic Exposure by means of books for a curious layperson
    • General Discussion
Replies
3
Views
367
A Why are General relativity texts so much more formal?
    • Special and General Relativity
Replies
28
Views
2K
The Math of Physics type books? (Not a methods book)
    • Science and Math Textbooks
Replies
4
Views
5K
  • Poll
Geometry A Comprehensive Introduction to Differential Geometry series by Spivak
    • Science and Math Textbooks
Replies
15
Views
19K
Holonomy of compact Ricci-flat Kaehler manifold
    • Differential Geometry
Replies
1
Views
3K

Hot Threads

Recent Insights

Back
Top