(anti)-symmetries of the Riemann curvature tensor

In summary, the Riemannian curvature tensor has four symmetries: (a) Rijkl=-Rjikl, (b) Rijkl=-Rijlk, (c) Rijkl=Rklij, and (d) Rijkl+Rjkil+Rkijl=0. Using these identities, it can be deduced that Rijmm=0 (summation over m implied). This is proved by using identity (b) and the fact that the metric is symmetric.
  • #1
quasar987
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The Riemannian curvature tensor has the following symmetries:
(a) Rijkl=-Rjikl
(b) Rijkl=-Rijlk
(c) Rijkl=Rklij
(d) Rijkl+Rjkil+Rkijl=0

This is surely trivial, but I do not see how to prove that

Rijkl=-Rjilk.

:(

Thanks.
 
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  • #2
This is supposed to be obvious, according to a commentary in a book.
I can get

Rijkl=-Rjilk

(from memory) but not the required identity...
 
  • #3
Contravariant and covariant indices must match on both sides of an equality (else the left and right hand sides do not transform the same way), so the original identity makes no sense. The identity in post #2 is as close as you can get.
 
  • #4
True!

So do you know how we can deduce from identities (a)-(d) that

Rijmm=0

(summation over m implied)?
 
  • #5
quasar987 said:
True!

So do you know how we can deduce from identities (a)-(d) that

Rijmm=0

(summation over m implied)?

Yes, use (b). Specifically use the fact that the metric is symmetric to show that

[tex]{R_{ijm}}^m = g^{mn} R_{ijmn} = - {R_{ijm}}^m . [/tex]
 
  • #6
mh, I see! Thanks fzero.
 

1. What is the Riemann curvature tensor?

The Riemann curvature tensor is a mathematical object that describes the curvature of a space in terms of its metric. It is used in the field of differential geometry to study the geometric properties of spaces, such as their shape and size.

2. What do (anti)-symmetries of the Riemann curvature tensor mean?

(Anti)-symmetries of the Riemann curvature tensor refer to certain mathematical properties of the tensor. These properties dictate how the tensor behaves under certain transformations, such as rotations or reflections. They are important in understanding the curvature of a space and making calculations using the tensor.

3. How are (anti)-symmetries of the Riemann curvature tensor related to curvature?

The (anti)-symmetries of the Riemann curvature tensor are directly related to the curvature of a space. They allow us to mathematically express the curvature in terms of the components of the tensor, which provides a deeper understanding of the geometric properties of a space.

4. Can (anti)-symmetries of the Riemann curvature tensor be used to classify spaces?

Yes, the (anti)-symmetries of the Riemann curvature tensor can be used to classify spaces into different types based on their curvature properties. This classification is used in the study of differential geometry and has applications in various fields, such as physics and cosmology.

5. Are there any real-world applications of (anti)-symmetries of the Riemann curvature tensor?

Yes, the (anti)-symmetries of the Riemann curvature tensor have many real-world applications. They are used in the study of general relativity, which describes the behavior of gravity in our universe. They also have applications in computer graphics, robotics, and computer vision, among others.

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