Find # of Independent Components of Riemann Curvature in D Dimensions

Click For Summary
SUMMARY

The number of independent components of the Riemann curvature tensor in D-dimensional space-time is calculated using the formula (1/12)*n^2*(n^2-1), where n represents the number of dimensions (n=D). This tensor is an irreducible representation of GL(4, ℝ) and adheres to the Bianchi identity R_{[\mu\nu|\rho]\lambda}=0. The solution can be found in Nakahara's text on pages 231-232, which provides a comprehensive explanation of this calculation.

PREREQUISITES
  • Understanding of Riemann curvature tensor and its properties
  • Familiarity with GL(4, ℝ) representations
  • Knowledge of Bianchi identities in differential geometry
  • Basic concepts of tensor calculus
NEXT STEPS
  • Study the derivation of the Riemann curvature tensor components
  • Explore the implications of Bianchi identities in general relativity
  • Review Nakahara's text for detailed examples and applications
  • Learn about irreducible representations in higher-dimensional geometry
USEFUL FOR

This discussion is beneficial for physicists, mathematicians, and students specializing in differential geometry, general relativity, or theoretical physics, particularly those interested in the properties of curvature tensors in various dimensions.

dextercioby
Science Advisor
Insights Author
Messages
13,410
Reaction score
4,207
How do i find the number of independent components of the Riemann curvature tensor in D space-time dimensions.

One is given that the Riemann tensor is an (2,2) irreducible rep of [itex]GL(4, \mathbb{R})[/itex] and obeys Bianchi I

[tex]R_{[\mu\nu|\rho]\lambda}=0[/tex]

Been trying this problem for 3 days and couldn't come up with a reasonable answer. :frown:


Daniel.
 
Physics news on Phys.org
(1/12)*n^2*(n^2-1) where n=D.
 
Tx, but i found it solved in Nakahara pages 231-232.

Daniel.
 

Similar threads

Showing that the Weyl tensor is invariant under conformal symmetries
  • · Replies 1 ·
Replies
1
Views
4K
Undergrad Reconciling 2 expressions for Riemann curvature tensor
  • · Replies 34 ·
2
Replies
34
Views
6K
R computation from 1 independent Riemann tensor component
  • · Replies 3 ·
Replies
3
Views
2K
What is the Riemann Curvature Tensor for Flat and Minkowski Space?
  • · Replies 8 ·
Replies
8
Views
2K
Find the Riemannian curvature tensor component
  • · Replies 6 ·
Replies
6
Views
2K
How many independent components does the Riemann curvature tensor have?
  • · Replies 16 ·
Replies
16
Views
7K
Undergrad Number of independent components of the Riemann tensor
  • · Replies 6 ·
Replies
6
Views
5K
Undergrad Ricci notations and visualisation
  • · Replies 10 ·
Replies
10
Views
2K
Undergrad Ricci curvatures determine Riemann curvatures in 3-dimension
  • · Replies 3 ·
Replies
3
Views
6K
Undergrad Null energy condition constrains the metric
  • · Replies 5 ·
Replies
5
Views
2K