# Tagged Questions

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### 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 ...
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### 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. ...
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### 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 ...
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### 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 $l\in L$ ...
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 ...