Does Metric Signature Affect Torsion Definition?

Click For Summary
SUMMARY

The discussion centers on the differing definitions of the torsion tensor, denoted as ##T^{c}_{ab}##, in two sources. One source defines it as ##T^{c}_{ab}=-\Gamma^{c}_{ab}+\Gamma^{c}_{ba}##, while the other states it as ##T^{c}_{ab}=\Gamma^{c}_{ab}-\Gamma^{c}_{ba}##. The conversation highlights the importance of conventions in mathematical texts, particularly regarding the definitions of the Riemann and Ricci tensors. Understanding these definitions is crucial for accurate application in differential geometry.

PREREQUISITES
  • Familiarity with differential geometry concepts
  • Understanding of torsion tensors
  • Knowledge of connection coefficients, specifically ##\Gamma^{c}_{ab}##
  • Basic comprehension of Riemann and Ricci curvature tensors
NEXT STEPS
  • Research the definitions of torsion tensors in various mathematical texts
  • Examine the implications of different conventions in differential geometry
  • Study the relationship between torsion and curvature tensors
  • Explore advanced topics in Riemannian geometry and their applications
USEFUL FOR

Mathematicians, physicists, and students of differential geometry who seek to understand the nuances of torsion and curvature definitions in various mathematical contexts.

binbagsss
Messages
1,291
Reaction score
12
I'm looking at 2 sources.

One has it defined as ##T^{c}_{ab}=-\Gamma^{c}_{ab}+\Gamma^{c}_{ba}##
And the other has ##T^{c}_{ab}=\Gamma^{c}_{ab}-\Gamma^{c}_{ba}##

##T## the torsion tensor
and
##\Gamma^{c}_{ab}## the connection.

Or is it more that different texts use different conventions?

Thanks.
 
Last edited:
Physics news on Phys.org
What are those sources?
What are their definitions of Riemann and Ricci?
 

Similar threads

Graduate Index-Free Expressions & Unique Metric-Compatible Connections
  • · Replies 1 ·
Replies
1
Views
2K
Graduate Equivalent definitions of tensor field
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Christoffel's symbols: symmetry in the two (lower) indices
  • · Replies 6 ·
Replies
6
Views
6K
Undergrad Electric Field seen by an observer in motion
  • · Replies 59 ·
2
Replies
59
Views
8K
Graduate How to Express the Schwarzschild Metric in Fermi Normal Coordinates?
  • · Replies 13 ·
Replies
13
Views
2K
Graduate Exploring the Stress-Energy Tensor of a Perfect Fluid
  • · Replies 3 ·
Replies
3
Views
2K
Graduate Covering Group of SO(g) & Understanding Spinors on Curved Spacetime
  • · Replies 15 ·
Replies
15
Views
2K
Graduate Curvature with different connections on the two-sphere
  • · Replies 15 ·
Replies
15
Views
2K
Graduate Write Torsion Tensor: Definition, Metric Tensor & Equation
  • · Replies 1 ·
Replies
1
Views
2K
Graduate Curvature & Connection Without Metric
  • · Replies 1 ·
Replies
1
Views
2K