• Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.


New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback


(propose a free ad)

Site Statistics

191 submissions , 151 unreviewed
4,796 questions , 1,987 unanswered
5,284 answers , 22,472 comments
1,470 users with positive rep
773 active unimported users
More ...

  The Einstein-Kaehler equations

+ 2 like - 0 dislike

Let $(M,g,J)$ be a Kaehler manifold ($\nabla J=J \nabla$), let $R(X,Y)$ be the riemannian curvature. I define:

$Ricc(J)=\sum_i R(J e_i, e_i)$

for an orthonormal basis $(e_i)$

$R(J) = J Ricc (J)$

$r(J)=tr (R(J))$

then I can define the Einstein-Kaehler equations:

$R(J)_{ij} - (1/2) r(J) g_{ij} =T_{ij}$

Can I reformulate the gravitation by means of these equations?

asked Aug 19, 2018 in Theoretical Physics by Antoine Balan (475 points) [ revision history ]
edited Aug 21, 2018 by Antoine Balan
If I am not mistaken, your Ricc(J) is the Ricci curvature 2-form (obtained from the usual Ricci curvature as the Kähler form is obtained from the metric). In particular, R(J)_ij is antisymmetric in ij, whereas g_ij and T_ij are symmetric in ij so your equation seems trivial. To obtain a reformulation of Einstein equation, you need to replace g_ij by \omega_ij (and T_ij by the associated 2-form).

Ricc(J) isn't just obtained from the usual Ricci curvature, moreover it is antisymmetric, so that R(J) is symmetric.

For an hermitian metric, but not necessary complex (almost-complex), I propose to take:


Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
To avoid this verification in future, please log in or register.

user contributions licensed under cc by-sa 3.0 with attribution required

Your rights