I'm trying to understand how to compute the null future-directed vector fields if I have a given (Bondi) metric $g=-e^{2\nu}du^{2}-2e^{\nu+\lambda}dudr+r^{2}d\Omega$ with $d\Omeg …

Let $(M,g)$ be a Riemannian manifold. Is it true that for any symmetric 2-tensor $\alpha$ we have: $Trace_g(\alpha)=1/\omega_n\int_{S^{n-1}}\alpha(V,V)dvol(V)$ where $w_n$ is the v …

I'm trying to prove the energy momentum tensor in curved spacetime for Electromagnetic field is Divergence-less directly(Without using general lie derivative method which can prove …

Hi I know how to calculate some easy tensor products like $\mathbb{Z}/m\mathbb{Z} \otimes_{\mathbb{Z}} \mathbb{Z}/n\mathbb{Z}\cong_{\mathbb{Z}} \mathbb{Z}/(m,n)\mathbb{Z} $ or $F[ …

The notation for tensors is like the plumbing in a very old Vermont farmhouse. It may once have been intentionally designed, but after that it just evolved. As an example, it seems …

I am puzzled by the amazing utility and therefore ubiquity of two-dimensional matrices in comparison to the relative paucity of multidimensional arrays of numbers, hypermatrices. O …

I am using usual physics notations and I guess the physics motivations of this question are obvious. Let a basis of the $SO(n,m)$ Lie algebra be denoted by $S^{\mu \nu}$ and the …

Let $M$ be a smooth, pseudo-Riemannian manifold with $\dim(M) \ge 2.$ Let $\nabla$ be any affine connection on $M$. No reason for it to be the Levi-Civita connection. All we assume …

Green-St Venant strain tensor is defined by $E(u)={1\over 2}[\nabla u+(\nabla u)^T+(\nabla u)^T\nabla u]$, where $\nabla u$ is the displacement gradient. Show that $u\in H^1(\Om …

Is there a simple example that shows that the functor $B\otimes_R(-)$ is not left exact, given a ring $R$ and a right $R$-module $B$?

Who coined the name "tensor" and why? What does the word "tensor" really mean, not the mathematical definition?

The covariant differentiation or the levi civita connection represented by the Christoffel symbol, $$\Gamma_{\alpha,\beta}^{\gamma} \frac{dx^k}{dy^{\gamma}} =?= \Gamma_{i,j} \fra …

Let $V$ be a vector bundle over a smooth, complete Riemannian manifold $M$. In general, the manifold is not compact. Further, denote by $g$ the metric on $M$, the volume measure b …

1)Why embedding of ( not necessarily finite-dimensional) vector spaces $V\rightarrow W$ produces embedding of tensor algebras $T(V)\rightarrow T(W)$. I can prove it using Hamel ba …

Please excuse me if this question turns out to be incredibly silly for one reason or another. Are there tensor/hypermatrix analogues of $GL(n,\mathbb{C})$ that are interesting? Wh …