6
votes
1answer
257 views

Equivalence of exterior forms

Let us start with the following definition. Let $1\leqslant k\leqslant n$ and let $\omega_1,\omega_2\in\Lambda^k(\mathbb{R}^n)$. We say that $\omega_1$, $\omega_2$ are equivalent, if there exists ...
1
vote
1answer
369 views

A multilinear question and its smooth version

Let $E$ and $F$ be two finite dimensional vector spaces. For every $k\in \mathbb{N}$, $E^{k}$ has a natural vector space structure and is isomorphic to $E\otimes \mathbb{R}^{k}$, in a natural way. ...
3
votes
2answers
278 views

Relationship between curvature tensor, algebraic Bianchi identity and sectional curvature

I am currently trying to understand the algebraic Bianchi identity, and I am clearly missing some purely algebraic fact. Let $M$ be a Riemannian manifold, $R$ its curvature tensor (with index ...
0
votes
0answers
122 views

When a hyperplane of symmetric forms is determined by a quadric hypersurface?

Let $L$ be a 2D real vector space, $L^*$ its dual, and $\{V,\omega\}$ the symplectic space with $V=L\oplus L^*$ and $\omega$ unambiguously defined by $\omega(l,\lambda):=\lambda(l)$, for all $l\in L$ ...
1
vote
0answers
274 views

Tensors as multilinear maps

I am aware that many books on differential geometry define tensors as multilinear maps. Namely $$ V\otimes W := L_2(V^* \times W^*,\Bbb F) $$ I am also aware that this space is isomorphic to the ...
3
votes
2answers
253 views

An expression with an alternating trilinear form, written in terms of the determinant and a symmetric bilinear form

I am trying to understand a line from MacLachlan/Reid's The Arithmetic of Hyperbolic 3-Manifolds (it's in 3.4 if you have the book), that seems it should be elementary but I can't seem to find where ...