All Questions
Tagged with tensor riemannian-geometry
25 questions
4
votes
1
answer
547
views
Question on Lorentzian geometry
I apologize in advance if this is a too basic question.
Let $(M,g)$ be a Lorentzian manifold with signature convention $(-,+,\dots,+)$. Now, lets suppose $X\in\Gamma(TM)$ defines a global time-...
- 1,171
5
votes
1
answer
464
views
Bochner Laplacian in coordinates
Sorry if this is a too basic question, but I didn't find an answer anywhere:
The connection Laplacian, or Bochner Laplacian, is the differential operator acting on $k$-tensor fields $T\in\Gamma^{\...
- 1,171
1
vote
0
answers
315
views
About "residual" scalar curvature in Einstein warped product manifold
I have an Einstein warped product manifold $M=B \times_f F$ with warped metric $g=g^B+f^2g^F$, where $F$ is Ricci flat, then its scalar curvature is $S^F=0$.
It is well known that the scalar curvature ...
- 272
2
votes
0
answers
44
views
$1$-parameter family of metrics preserving the normal direction
Let $(M^n,g)$ be a compact Riemannian manifold with boundary, $n \geq 2$, and let $N$ be the unit outward normal to $\partial M$. Denote by $S^2(M)$ the symmetric covariant $2$-tensors, by $S^2_0(M)$ ...
- 2,095
1
vote
1
answer
206
views
Decomposition of tensor field on hypersurface
Let $(\mathcal{M},g)$ be a Lorentzian manifold, which is globally of the form $\mathcal{M}\cong I\times\Sigma$, where $I\subset\mathbb{R}$ ("time") and $\Sigma$ ("space") is some $...
- 1,171
5
votes
2
answers
339
views
Example of a curvature with no associated metric
Is there a concrete example of a $4$ tensor $R_{ijkl}$ with the same symmetries as the Riemannian curvature tensor, i.e.
\begin{gather*}
R_{ijkl} = - R_{ijlk},\quad R_{ijkl} = R_{jikl},\quad R_{ijkl} =...
- 113
6
votes
1
answer
766
views
Does every ‘curvature’ tensor induce a metric? [duplicate]
So we know that given a Riemannian manifold $(M,g)$ we can calculate its Riemannian curvature $R$. This covariant $4$-tensor field then satisfies some important symmetries
\begin{gather*}
R_{ijkl} = - ...
- 113
5
votes
2
answers
572
views
Bounds for metric in normal coordinate
Let $M$ be a Riemannian $n$-manifold and $x \in M$. For the metric tensor $g_{ij}$ of geodesic normal coordinates at $x$, there is a formula $g_{ij}(y) = \delta_{ij} + \frac13 R_{kijl} y^k y^l + O(\|y\...
- 903
5
votes
2
answers
425
views
Local diagonalisation of a degenerated 2d metric tensor
Consider a smooth 2d-manifold $M$ and let $g$ be a smooth $(0,2)$-tensor satisfying $rk(g)\geq1$ everywhere. Obviously if $rk(g)=2$ at a point $p\in M$ then $g$ is locally diagonalisable (i.e. there ...
- 275
2
votes
1
answer
298
views
Decomposition of tensors
It is well known that every traceless, symmetric $2$-tensor can be decomposed uniquely into a Lie derivative part and a Codazzi part. Is there an analog for totally symmetric $k$-tensors?
- 38
2
votes
1
answer
1k
views
Is there a geometric intepretation of the trace of tensor on a Riemannian manifold?
For a long time I thought the trace for matrices were just an elementary function with nice properties. But it is much more than that and really should be think of as a geometrical object (see ...
- 4,361
5
votes
0
answers
101
views
How is this product of tensors defined?
I am reading the paper “ The first eigenvalue of a small geodesic ball in a riemannian manifold”, by Karp and Pinsky, from where I took the following:
Here, $\Delta_{-2}$ denotes the usual Laplacian ...
- 2,095
1
vote
0
answers
138
views
References Request: Bach tensor
Recently I want to study about Bach tensor in detail. From the fundemental definition and properties to conformal invariant. Is there any references to me about Bach tensor? If there are some origins ...
- 11
2
votes
1
answer
137
views
Mean Gaussian curvature from non-unit vector
Pg.248 of "Textbook in Tensor Calculus and Differential Geometry" by Prasun Nayak.
Let us suppose that $\lambda_{h|}^i$
is not a unit vector and therefore, the mean curvature $M_h$ in
this ...
- 23
-1
votes
1
answer
526
views
Christoffel symbols as the expansion coefficients of covariant/contravariant derivatives
Page 155 of Vector and Tensor Analysis with Applications, by A.I Borishenko and I.E. Tarapov, the authors state that Christoffel symbols of the second kind are expansions of $\frac{\partial {\bf e}_j}{...
- 661
8
votes
1
answer
238
views
Question on Nash's paper on $C^1$ isometric immersions: Why approximating the error tensor $\delta$?
I am trying to go through the classical paper by Nash on the existance of $C^1$ isometric immersion of a Riemannian manifold $(M,g)$ (here is the Jstor link: https://www.jstor.org/stable/1969840?seq=1#...
- 203
1
vote
1
answer
142
views
Connection of the existence of Killing-Yano tensor and Killing tensor
Stephani states that in 4 dimensions a spacetime admits a non-reducible Killing-Yano tensor only if the Weyl tensor either is
of Petrov type D or vanishes. Does this imply that the spacetime also ...
- 679
1
vote
0
answers
922
views
On a Riemannian manifold, calculate the metric from the distance [closed]
Given a Riemannian manifold $(M,g)$ is it possible to calculate the distance between two points on this manifold. Is it possible the inverse? That means: given a formula of the distance, for example:
...
- 399
2
votes
2
answers
515
views
A kind of "Curvature tensor" for higher dimensional tensors
I begin my question with a multilinear question then I will consider two local smooth analogies:
Assume that $\alpha$ is a real valued symmetric $k$-tensor, that is a $k$-linear map $\alpha:\...
- 356
2
votes
1
answer
194
views
Vanishing of determinant of Cotton York tensor
suppose $\Omega \subset \mathbb{R}^3$ is a Riemannian manifold with $g= dx_1^2 + dx_2^2 + c(x_1,x_2,x_3)dx_3^2$. Is it true that $det(CY)=0$?
Thanks
- 4,135
3
votes
0
answers
367
views
Obtaining the metric from the mixed Ricci tensor $R^i{}_j$
In chapter 5 of the book "Einstein Manifolds", Arthur Besse discusses the possibility to find the metric $g$ when knowing the Ricci curvature tensor $Ric(g)$ ($=R_{ij}$).
But what do we know about ...
- 4,312
2
votes
2
answers
5k
views
Calculating the Riemann Christoffel tensor for a diagonal metric
I am trying to calculate the entries of the Riemann curvature tensor $R^m_{\phantom{m}ijk}$ for the metric $g_{ij}$.
The Riemann-Christoffel tensor is given as
\begin{align}
R^m_{\phantom{m}ijk} = \...
- 219
4
votes
1
answer
747
views
The Laplacian of an expression involving the Ricci tensor
While doing some computations on a compact Riemannian manifold I have reached the following expression:
$$ \Delta_y \big( Ric_y (\exp_y ^{-1} x, \exp_y ^{-1} x) \big) (x)$$
where $\Delta_y$ is the ...
- 5,407
19
votes
4
answers
3k
views
Equations satisfied by the Riemann curvature tensor
It is well known that the Riemann curvature tensor of a metric satisfies
\begin{eqnarray}
R_{jikl}=-R_{ijkl}=R_{ijlk},(1)\\
R_{klij}=R_{ijkl},(2)\\
R_{i[jkl]}=0 \mbox{(1st Bianchi identity)}.(3)
\end{...
- 21.8k
1
vote
1
answer
250
views
Global geometry measures for Riemannian manifolds
I'm working on a stochastic algorithm and considering it to apply in case of any curved space (manifolds). But in order to make the algorithm as efficient as possible I want to include in it some ...
- 267