Tagged Questions
2
votes
1answer
36 views
Contraction of the second Bianchi identity
The second Bianchi identity is
$${R^a}_{b[cd;e]}=0$$
And contracting it with respect to $a$ and $e$ we get
$${R^a}_{b[cd;a]}=0 \Leftrightarrow $$
$${R^a}_{bcd;a}+R_{bc;d}-R_{bd;c}=0$$
What I don't ...
1
vote
1answer
71 views
Quotient theorem for tensors
Can somebody please explain to me how the following statement is true?
The Riemann curvature tensor $R^c_{dab}$ is given by the Ricci identity $$(\nabla_a\nabla_b-\nabla_b\nabla_a)V^c\equiv ...
1
vote
0answers
63 views
Riemannian curvature and its application on covariant derivative of tensors
This identity can be generalized to get the commutators for two
covariant derivatives of arbitrary tensors as follows $ \begin{align} T^{\alpha_1 \cdots \alpha_r}{}_{\beta_1 \cdots \beta_s ; ...
0
votes
0answers
75 views
Tensor calculus solution-why?
The text I read says that $\displaystyle\frac{\partial^2
x^\alpha}{\partial x^\delta
\partial x^\gamma}\frac{\partial x^\delta}{\partial
x^\beta} = 0$ leads to the solution $x^\alpha =
...
1
vote
0answers
149 views
How to derive covariant derivative and Lie derivative of tensors
1) As title says, how does one derive the following
equation for covariant derivate of tensor:
$A^{\alpha}{}_{;\beta} = A^{\alpha}{}_{,\beta} +
\Gamma^{\alpha} {}_{\gamma\beta}A^\gamma$
where ...
1
vote
0answers
43 views
Tensor Components
I would like to ask something
On the Barrett Oneill's Semi-Riemann Geometry there is a definition of tensor component:
Let $\xi=(x^1,\dots ,x^n)$ be a coordinate system on $\upsilon\subset M$. If $A ...
3
votes
0answers
84 views
What are spinor fields?
For example, a tensor field $T=T_{a,b}^{\ \ \ \ c}$ on a manifold $M$ should be thought as
$$
T_{a,b}^{\ \ \ \ c}dx^{a}\otimes dx^{b}\otimes \frac{\partial}{\partial x^{c}}
$$
with local coordinate ...
2
votes
1answer
39 views
Question on tensor calculation in Reimannian geometry
Given a Riemannian manfiold $M$ with metric $g=(g_{i,j})$. Let $T=T_{A,B,C,\dots}^{a,b,c,\dots}$ be a tensor on $M$. I would like to compute for example $T_{A,B,C,\dots}^{a,n,c,\dots}g_{n,m}$. We ...
3
votes
2answers
195 views
Is there such a thing as discrete Riemannian geometry?
General relativity expresses gravity as a curvature in space time, created by the stress energy tensor.
$$T_{\mu\nu} \approx R_{\mu\nu} - \frac{R}{2} g_{\mu\nu}$$
Given I put the fact that energy is ...
2
votes
0answers
109 views
Extending Tensor Fields defined on Manifolds to Ambient Space
I am currently reading about tensor fields on manifolds, and I came across two comments that sound contradictory to me.
The first comment is made in the book by James Munkres "Analysis on Manifolds", ...
2
votes
1answer
105 views
Simple problem with the normal curvature tensor
If $M$ is a s-R(semi-Riemannian) submanifold of a s-R manifold $\overline{M}$ the function $R^{\perp}:\mathfrak{X}(M)\times\mathfrak{X}(M)\times\mathfrak{X}(M)^\perp\rightarrow\mathfrak{X}(M)^\perp$ ...
