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3

Hint the determinant is continuous so the inverse image of the real line -0 by the determinant is an open subset so it is a submanifold.


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Before saying what I think are good introductions to mathematical gauge theory, I should say what I think gauge theory is. What gauge theory means to me is the application of certain PDEs, relevant in physics, to the topology and geometry of manifolds. I note here that one can be ignorant of the actual physics (as I am). Here is one example. Given a ...


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You're doing great so far. Let me use the name $\eta$ for the thing you call $\theta$, and then continue. You need to integrate $\eta$ over something that projects to the unit circle in the $xy$ plane. So let's take $$ \gamma(t) = (\cos t, \sin t, f(\cos t, \sin t)) $$ which parameterizes the set $\partial R_f$ nicely, and use it to compute a path ...


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The answer is yes for the following reason: $\omega_n$ is the Euler class mod 2 Now use e.g. Theorem 4.7 here which says $e(\nu_N)([N])$ counts number of intersections, or you argue that the Thom class of $\nu_N$ in $M$ is the Poincaré dual of $N$ (follows from this exercise). Hence, by pulling back to the cohomology of $N$ the result follows. You ...


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$\newcommand{\Reals}{\mathbf{R}}\newcommand{\Brak}[1]{\langle #1 \rangle}$Strictly speaking, manifolds and vector spaces are "different types of fruit": Sets equipped with extra structure in such a way that neither structure naturally determines the other. Further, the term isometry gets used without qualification in multiple settings, including (but ...


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To show that the Jacobian really is of rank $1$ (i.e., not equal to the zero vector), I suggest you expand $\det A$ by minors along the first column, say: $$\det A = \sum_{i=1}^n (-1)^{i+1}a_{i,1}M_{i,1}.$$ Of course, $(-1)^{i+1}M_{i,1}$ is in fact an entry of the Jacobian. So the Jacobian can only be zero, if also $\det A=0$. But we are looking at the ...


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In other words, you are asking if there are "exotic" coequalisers in the category of manifolds. In fact, there are. Let $S^1 = \{ z \in \mathbb{C} : \left| z \right| = 1 \}$ be the circle, let $R = \mathbb{Z} \times S^1$, and let $d_0, d_1 : R \to S^1$ be defined as follows: $$d_0 (n, z) = \exp (i n) z$$ $$d_1 (n, z) = z$$ It is not hard to see that the ...


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In the meantime I have come up with a similar answer to Zhen Lin's. Consider $\mathbb{Q}$ with the discrete topology, then the colimit of the coproduct map $\mathbb{Q} \rightrightarrows \mathbb{R} \coprod \mathbb{R}$, where the two arrows are the two obvious inclusions, has the colimit $$ \begin{array}{rcl} \mathbb{R} \coprod \mathbb{R} & \to & ...


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The key to answering your questions in my opinion is to get a nice description of densities in local coordinates. (For this purpose, the definition of densities that you have chosen does not seem very convenient to me). The point is that, as for $n$-forms, one can describe the restriction of a density to the domain of a coordinate chart by a single smooth ...


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$\newcommand{\Reals}{\mathbf{R}}\newcommand{\Basis}{\mathbf{e}}\newcommand{\Cpx}{\mathbf{C}}\newcommand{\Proj}{\mathbf{P}}$I'm guessing your notation and terminology come from physics. To forestall possible confusion, I'm using mathematical notation and terminology: The unit ball $B^{n} \subset \Reals^{n}$ is the set of points lying at most one unit from ...


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It's better to express $\frac{\partial }{\partial x}, \frac{\partial }{\partial y}, \frac{\partial }{\partial z}$ in terms of $\frac{\partial }{\partial u}$, $\frac{\partial }{\partial v}$: $$\frac{\partial }{\partial x}=\frac{1}{1-z}\frac{\partial }{\partial u}\\ \frac{\partial }{\partial y}=\frac{1}{1-z}\frac{\partial }{\partial v}\\ \frac{\partial ...


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Examining the Gauss map, we see that $\overline{g}$ is covered by the bundle map$$g \oplus g^\perp: \tau \oplus \nu \to \gamma_n \oplus \gamma^\perp$$and the differential$$D\overline{g}: \tau M \to \tau G_n(\mathbb{R}^{n+k}) \simeq \text{Hom}(\gamma^n, \gamma^\perp).$$This thus yields a fiber-linear morphism$$\sigma: \tau M \to \text{Hom}(\tau M, ...



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