# Tagged Questions

Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such ...

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### Defining a Riemannian metric

Let $M$ be a smooth manifold. I have seen a Riemannian metric be defined in many ways: A smooth choice of an inner product $g_p:T_pM\times T_pM\to\mathbb{R}$ which is symmetric and positive-definite,...
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### Reference about the surgery of Ricci flow

I roughly read the Topping's LECTURES ON THE RICCI FLOW. Seemly, there is not introduction about surgery. Seemly,it is enough to deal singularity by blow up. Then, for knowing surgery I read the ...
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### Pullback Courant algebroid

I am reading the lecture notes on generalized geometry http://www.staff.science.uu.nl/~caval101/homepage/Research_files/australia.pdf but now I am stuck on the exercise 1.31 at page 13. Here is the ...
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### Do these assumptions on a mapping ensure it is a diffeomorphism?

$A\subset \mathbb{R}^m$ is an open and bounded set, $f:A\longrightarrow \mathbb{R}^n$, $m\leq n$, is injective, continuously differentiable and its Jacobian matrix has full rank on $A$. Does this ...
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### Is a surjective mapping of R2 to itself with full rank derivative everywhere necessarily injective?

If $f:\mathbb R^2\rightarrow\mathbb R^2$ has rank 2 derivative everywhere, then by the inverse function theorem it is locally injective. If it is surjective, is it then necessarily globally injective ...
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### Unique Solution to Equation in Two Variables & Possible Use of the Implicit Function Theorem

Let $g(x) : R \to R$ be a continuous function; Consider the equation $T(x,y) = y^3 -y^2 +(1+x^2)y - g(x)$ Show that for a given $x$ there exist a unique solution $y$ to the equation $T(x,y) = 0$. ...
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### How is “expressing” a differential operator “in cylindrical coordinates” rigorously defined?

I'm a mathematician (with little knowledge of differential geometry) trying to study physics. One of the greatest problems is the language regarding coordinate transformations. I tend to think of such ...
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### tangent space in a moving coordinate frame

I've got a problem in some geometry of flow. For the sake of completeness I will give the complete derivation of the equation of interest, but I will seperate it into derivation part and question ...
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### Cartan decomposition diffeomorphism at the level of (compact) groups

I am trying to understand the Cartan decomposition in the case of a compact group $G$ with subgroup $K$, such that their respective Lie algebras $(\mathfrak{g},\mathfrak{k})$ correspond to a Cartan ...
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### Is integration with respect to spherical measure equivalent to manifold integration over sphere?

Let $S$ be an $n$--sphere $\mathcal{S}^n(0,R)\subset\mathbb{R}^{n+1}$, $\Theta\subset\mathbb{R}^n$ an open subset and let $\phi:\Theta\subset\mathbb{R}^n\longrightarrow S\subset\mathbb{R}^{n+1}$ be a ...
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### Planar intersections of constant Gauss curvature K surfaces

Have they been studied? It appears they did not generate enough interest except the conic sections $K=0$. Do they give rise to curves of fourth order?
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### About transformations of the metric: should we use the old or the new one to raise/lower indices?

Let $(M,g)$ be a (Pseudo-)Riemannian manifold. If I perform a transformation on the metric, getting a new metric $\tilde{g}$, which metric should I use to raise and lower indices? As I understand, ...
Asking about a step regarding indices in deriving the Curvature tensor from the geodesic equation. Starting from $$\frac{d v^a}{du} = - \Gamma^a_{bc}v^b \frac{dx^c}{du}$$ we integrate v^a(u) = ...
### I want to show that $\mid D pr(m) \mid \leq \dfrac{1}{1-\Vert pr(m)-m \Vert \Vert h_{pr(m)}\Vert}.$
Let M be a smooth surface, and $U$ a neighborhood where the orthogonal projection pr is well defined. I would like to show that $\forall m \in U$, if $m \in S$, pr is differentiable on m and we ...