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Action of Lie group on vector fields (of homogeneous space)

1) If I have a Lie group $G$ acting smoothly on a manifold $M$, how does this gives also an action of $G$ on the vector fields on $M$? 2) When I read that the group G acts canonically on the vector ...
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Schubert cell decomposition and full flags

I am looking for a self-contained basic theory of Schubert-cells through finding the decomposition of the full flag $Fl_3(\mathbb C^3)$.
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low-dim unitary groups and their actions

I need someone to explain for me the unitary groups $U(1)$, $U(2)$ and $U(3)$ and their actions: Specifically: $U(3)/U(2)$ $U(3)/U(2)\times U(1)$ $U(3)/U(1)\times U(1) \times U(1)$ I have seen ...
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2D Homogeneous Transformation : Reflection vs Mirroring

I have two questions: (1) Is there any difference between the terms Reflection and Mirroring in 2D Transformation? (2) What are their Transformation matrices with reference to an arbitrary line?
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Extracting sign of scaling from modelView matrix

I want to retrieve the sign of scaling for each axis from modelview matrix. Right now I am able to extract the sign only if all 3 signs are same but it fails when one of them is different. Here is the ...
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Orbits of $SL(3, \mathbb{C})/B$

Let $B= \Bigg\{\begin{bmatrix} * & *&* \\ 0 & *&*\\ 0&0&* \end{bmatrix} \Bigg\}< SL(3,\mathbb C)$. What is $SL(3,\mathbb C)/B$? Do we use these facts: Borel fixed ...
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Orbit under Lie group and projective variety

On https://en.wikipedia.org/wiki/Generalized_flag_variety#Highest_weight_orbits_and_homogeneous_projective_varieties there is a section which says Blockquote If G is a semisimple algebraic group ...
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Metric of Lorentzian signature on a compact homogeneous space.

Can anyone give an example of a metric of Lorentzian signature on a compact homogeneous space.
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Are Homogenous countable complete metric spaces always discrete?

Let $M$ be a countable complete metric space such that the group of isometries of $M$, $Iso(M)$ acts transitively on the points in $M$. Does it follow that the topology induced by the metric is the ...
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What is the dimension of the space of planes in $\Bbb R^3$?

What is the dimension of the space of planes in $\Bbb R^3$ and how do we reach the answer? Clarification: What I am searching for is what is the least number of parameters that I need. For example, ...
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