## Riemann-Tensor have in n- dimensional space?

How many independant components does the Riemann-Tensor have in n- dimensional space?

 PhysOrg.com science news on PhysOrg.com >> Heat-related deaths in Manhattan projected to rise>> Dire outlook despite global warming 'pause': study>> Sea level influenced tropical climate during the last ice age
 Blog Entries: 47 Recognitions: Gold Member Homework Help Science Advisor

 Quote by Thaakisfox How many independant components does the Riemann-Tensor have in n- dimensional space?
this is related to the number of components related to riemanian tensor and the space that contains it
as a simple example if u have a mixed tensor of 5 components 3 contarvariant and 2 covariant in 4 dimensional space so u have 4^5 compnents i.e 1024
u see how many equations are contracted to single one that's why einstein begin his general relativity by studying tensors with his friend Grassmann

my name is mina, i study QFT

## Riemann-Tensor have in n- dimensional space?

 Quote by Thaakisfox How many independant components does the Riemann-Tensor have in n- dimensional space?
for more information see Shaum vector analysis chap8

Recognitions:
 Quote by Thaakisfox How many independant components does the Riemann-Tensor have in n- dimensional space?
Just thought I'd stress that you are probably asking about the number of algebraically independent components. To understand the physical significance of a "geometric" field equation such as the Einstein field equation, you also need to appreciate a crucial differential relation, the (differential) Bianchi identity, which is crucial to understanding, for example, how it can happen in gtr that fluid motion inside some fluid filled region can give rise to gravitational radiation which propagatges as a wave across a vacuum region.

Recognitions:
Also, the example you gave overlooks the possibility of algebraic symmetries which will in general reduce the number of algebraically independent components. For example the Riemann tensor (more or less by definition) satisfies $R_{abcd} = -R_{bacd}$.