# Tagged Questions

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### Curve in union of hyperplanes

If a smooth curve $\gamma: [0,T] \to \mathbb R^n$ is contained in the union of hyperplanes $$\bigcup_{i=1, \dots, N} H_i$$ does it then follow that one can always find time intervals $[t_0, t_1]$ ...
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### normal vectors in spaces where $n > 3$

I am reading Lovelock and Rund's book on Tensors and they make a statement that I wanted to validate about normal vectors in high-dimensional spaces. It should be remarked that the above ...
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### The level set of a smooth function

Let $f$ be a smooth function on a manifold $M$. Fix a point $p\in M$ and let $df\in T^\ast_pM$ be the differential of $f$ at $p$. I read that the subspace of $T_pM$ of vectors $X$ such that $df(X)=0$ ...
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### set of almost complex structures on $\mathbb R^4$ as two disjoint spheres

The set of almost complex structures on $\mathbb R^{2n}$ is given by $$M_n = \frac{GL(2n,\mathbb R)}{GL(n,\mathbb C)} = \mathcal C_+ \sqcup \mathcal C_-,$$ taking into account that $\det = \pm 1$ ...
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### Distorted Unitary matrices

Let $U$ be an unitary and $D$ be a diagonal matrix. We know that for all vectors $v$ on the sphere $Uv$ is on the sphere and, $$\langle Uv,Uv\rangle=\langle v,v\rangle.$$ What are the vectors $v$ on ...
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### Grassmannian, Plucker coordinates

In which books can I find something about the grassmannian and the plucker coordinates ?
Please look at this problem: Let $\mathcal{H}$ be the space of $(n,n)$ hermitian matrix. \phi:\begin{align*} &\mathcal{H} \to \mathfrak{u}(n):=Lie(U(n)) \\&A \mapsto iA \end{align*} ...
I am trying to show the following equivalence: Proposition 1. A subspace $V$ of the Minkowski space $\mathbb{R}_1^{n+1}$ is non-degenerate if and only if $\langle v, v \rangle \ne 0$ for all \$v \in ...