Riemann tensor, Ricci tensor of a 3 sphere

In summary, the Christoffel symbols are:
  • #1
dingo_d
211
0

Homework Statement


I have the metric of a three sphere:

[itex]g_{\mu \nu} =
\begin{pmatrix}
1 & 0 & 0 \\
0 & r^2 & 0 \\
0 & 0 & r^2\sin^2\theta
\end{pmatrix}[/itex]

Find Riemann tensor, Ricci tensor and Ricci scalar for the given metric.

Homework Equations



I have all the formulas I need, and I calculated the necessary Christoffel symbols, by hand and by mathematica and they match. There are 9 non vanishing Christoffel symbols. Some I calculated and for others I used the symmetry properties and the fact that the metric is diagonal (which simplifies things).

But when I go and try to calculate Riemman tensor via:

[itex]R^{a}_{bcd}=\partial_d \Gamma^a_{bc}-\partial_c\Gamma^a_{bd}+\Gamma^m_{bc}\Gamma^a_{dm}-\Gamma^m_{bd}\Gamma^a_{cm}[/itex]

I get all zeroes for components :\

And I kinda doubt that every single component is zero.

The Christoffel symbols are:

[itex]
\begin{array}{ccc}
\Gamma _{\theta r}^{\theta } & = & \frac{1}{r} \\
\Gamma _{\phi r}^{\phi } & = & \frac{1}{r} \\
\Gamma _{r\theta }^{\theta } & = & \frac{1}{r} \\
\Gamma _{\theta \theta }^r & = & -r \\
\Gamma _{\phi \theta }^{\phi } & = & \cot (\theta ) \\
\Gamma _{r\phi }^{\phi } & = & \frac{1}{r} \\
\Gamma _{\theta \phi }^{\phi } & = & \cot (\theta ) \\
\Gamma _{\phi \phi }^r & = & -r \sin ^2(\theta ) \\
\Gamma _{\phi \phi }^{\theta } & = & -\cos (\theta ) \sin (\theta )
\end{array}
[/itex]
 
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  • #2
that is the metric of R^3 in sphereical coordinates with one of the angles constant. hense...a plane.
 
  • #3
Oh, so it's normal for Riemann tensor to be zero since there is no curvature. Phew, I thought I might be doing something wrong xD

Is there some easier way of showing that all of the components of Riemann tensor are zero, rather than manually calculating them all?

And how come when I look at 2-sphere

[itex]
g_{\mu \nu} =
\begin{pmatrix}
r^2 & 0 \\
0 & r^2\sin^2\theta
\end{pmatrix}[/itex]

I get several non vanishing components of Riemann tensor? Is it because I'm now looking it from my 3d perspective so I can see that the plane has to be curved to form a sphere?
 
  • #4
but now you can pull out r^2 which is clearly nonflat conformal.

you can't just throw away terms in the metric and think you're looking at an orthogonal projection onto the remaining coordinates.

if you are in a flat geometry in curvilinear coordinate systems then transforming the coordinate system back into flat coordinates might help ease the pain.
 
Last edited:
  • #5
What you have at the top is not the metric of a 3-sphere, but simply the metric of R^3 in spherical polar coordinates. A 3-sphere metric will have r fixed and 3 angles to describe where you are (a sphere is hollow remember), just as a 2-sphere has r fixed and 2 angles to describe where you are.
 

1. What is the Riemann tensor of a 3-sphere?

The Riemann tensor is a mathematical object that describes the curvature of a 3-sphere, which is a three-dimensional version of a sphere. It is a 4th-order tensor that contains information about the infinitesimal changes in direction of geodesics (curved paths) on the 3-sphere.

2. How is the Riemann tensor related to the curvature of a 3-sphere?

The Riemann tensor is directly related to the curvature of a 3-sphere. It is a measure of the non-Euclidean nature of the space and tells us how the shape of the 3-sphere differs from a flat space. A non-zero Riemann tensor indicates that the space is curved.

3. What is the Ricci tensor of a 3-sphere?

The Ricci tensor is a mathematical object that describes the local curvature of a 3-sphere. It is a 2nd-order tensor that is derived from the Riemann tensor and provides a more simplified description of the curvature of the space.

4. How is the Ricci tensor related to the Riemann tensor of a 3-sphere?

The Ricci tensor is derived from the Riemann tensor by contracting two of its indices. It is a more compact representation of the curvature information contained in the Riemann tensor. The Ricci tensor provides a more intuitive understanding of the curvature of a 3-sphere.

5. Why are the Riemann and Ricci tensors important in studying a 3-sphere?

The Riemann and Ricci tensors are important mathematical tools in the study of a 3-sphere because they allow us to quantify and understand the curvature of the space. By analyzing these tensors, we can gain insights into the properties of a 3-sphere and its geometry. They also play a crucial role in Einstein's theory of general relativity, which describes the relationship between gravity and the curvature of space.

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