5 votes

Stokes theorem for integrating a scalar times normal over a surface area

Classically we have two theorems in $\mathbb R^3$: Gauss' divergence theorem and Stokes' theorem. Your formula confuses me a bit as it seems to be neither one of them. A better notation is probably $$\...
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  • 4,169
5 votes
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Is a map which sends a $3\times 3$ symmetric tensor to an element of $SO(3)$ which diagonalizes it necessarily discontinuous?

Yes, discontinuities at repeated eigenvalues are essential for such a map. If $Q_0$ has a repeated eigenvalue and thus freedom to choose $R'(\theta)R,$ then for any choice of $\theta$ you can find $Q$ ...
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5 votes
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How to see that "two manifolds are diffeomorphic when you can give them each a coordinate atlas with the same transition maps"

Suppose that $X, Y$ are manifolds with atlases $\{\phi_\alpha: \alpha\in A\}$ and $\{\psi_\alpha: \alpha\in A\}$ such that for any two pairs of indices $\alpha, \beta\in A$, $$ \phi_\beta\circ \phi_\...
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4 votes
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Coordinate patches attached to a 2-manifold

These coordinate patches as seen in the picture are open sets $U_j\subset M$ subordinate to the atlas $(\varphi_i,U_j)$ of the manifold $M$ equipped with a coordinate grid that is induced by the ...
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  • 576
4 votes
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Utility of the coordinate free definition of the derivative on manifolds.

The importance comes in when you replace your spaces by manifolds which are not (open subsets of) $\mathbb{R}^n$. Then, you are by definition of a manifold always able to choose a local coordinate ...
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  • 1,086
4 votes

Utility of the coordinate free definition of the derivative on manifolds.

In addition to nictor000's answer, let me address your quarrels with Property 1. Indeed, there is no canonical map $T_pM\to M$, but there is something that takes its place. For any $v\in T_pM$ there ...
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  • 5,568
4 votes
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Smooth transition from Euclidean plane to hyperbolic plane

Yes, this is possible with a simple formula (although with an infinite family of spaces in between; see below). As $t \in [0,1]$ varies, just use the family of metrics $$e^{2ty} dx^2 + dy^2 $$ When $t=...
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4 votes

Convexity in "usual Partition of Unity arguments"

A convex subset of a vector space $C\subseteq V$ is a set which is closed under all convex combinations, which are wieghted sums whose wights are nonnegative and sum to $1$: $$ \lambda_1u_1+\...
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  • 13.1k
3 votes
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Lee Smooth Manifolds - Lemma (6.14) for Whitney Embedding Theorem

This point is slightly subtle (logically speaking). The idea is that Lee doesn’t need the exact value of $N$ at this point – it’s redefined as “some integer such that there is a smooth proper ...
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  • 27.2k
3 votes

Question about the covering space of a spin manifold

Suppose $\pi:\widetilde{M}\rightarrow M$ is a covering map of smooth manifolds and that $M$ is spin. This means what $w_1(TM) = w_2(TM) = 0$ where $w_i(TM)\in H^i(M;\mathbb{Z}/2\mathbb{Z})$ denote ...
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  • 45.1k
3 votes
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Dynamics on the torus

A rational slope of the orbit on the torus guarantees the orbit intersect and connect with itself, eventually. Thus the orbit must be diffeomorphic to $S^1$. Unwrap the blue orbit. See how it is ...
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3 votes

Dynamics on the torus

Here is a more rigorous argument (with gaps to fill in, depending on one's disposition). Denote by $\mathbb{T}^d=\mathbb{R}^d/\mathbb{Z}^d$ the (standard) $d$-torus and by $\operatorname{Aff}(\mathbb{...
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3 votes
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Is the curve $(\cos(t),\sin(t))$, for $t\in[0,2\pi[$, differentiable at $t=0$?

By (the most common) definition, differentiability is a local property at a point that requires an open neighborhood around the point, as you already pointed out. One weaker concept would be to ...
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  • 103
3 votes

How to find flow equation of a complex function?

The integral curves of the system $\dot x= u=x^2-y^2$ and $\dot y = v=2 xy$ can be found by first expressing the system in consolidated form as a single condition on complex quantities: the complex ...
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  • 1,472
3 votes

Push-forward of a smooth function

Welcome to MSE! I think you are right, actually, in some books the push forward is just called differential even if $N$ is not the real line and the push forward function is denoted by $df$. See even ...
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3 votes

How to see that "two manifolds are diffeomorphic when you can give them each a coordinate atlas with the same transition maps"

Consider two manifolds $M$ and $N$, with atlases $\{(U_{\alpha}, \phi_{\alpha})\}$ and $\{(V_{\alpha}, \psi_{\alpha})\}$ respectively. Let $\tau = \phi_{\beta} \circ\phi_{\alpha}^{-1}:\phi_{\alpha}(...
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2 votes

Is this Lie group isometric to the Euclidean plane?

The solution edited into the original post is brilliant. I'll just type a different way in which one could have found it (or rather: found the explicit isometry between $G$ and the plane). But in a ...
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  • 9,157
2 votes

Good applied differential geometry books

Personally I like differential geometry a lot and also like to find more about their applications. Not sure if there is an overview book talking about all the applications, but for each field there ...
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2 votes
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Rank of a coherent sheaf using resolution by vector bundles

Yes. Hilbert polynomials are additive over exact sequences: if $0\to F_n\to \cdots \to F_0\to 0$ is an exact sequence, then $\sum_{i=0}^n (-1)^iH_{F_i}(\lambda) =0$.
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2 votes

Foliations as equivalence relations

By definition, the leaves of a foliation is a partition of the manifold, and as any partition induces an equivalence relation, so does a foliation. See e.g. Anosov's entry for a foliation in the ...
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  • 6,834
2 votes
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What is a homothetic change of metric? How does it mean that using a homothetic change of metric we can set the volume equal to 1.

A homothetic change to the metric is simply a rescaling by a positive constant, i.e. replacing $g$ by $cg$ where $c \in (0, \infty)$. Now note that $d\mu_{cg} = c^{n/2}d\mu_g$. This can be seen in ...
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1 vote
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Kahler manifold computation

This answer is a more coherent explanation of the comments I wrote to the question. First, I think it might be important to note that a version of this Proposition holds in less restrictive situations ...
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1 vote
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Deriving with respect to arc length

$dy/dx = \tan \phi$ is the slope of tangent to a curve at any point. We interpret $ \sin \phi =dy/ds$ and $\cos \phi=dx/ds$, in the infinitesimal or differential triangle where instantaneous tangent ...
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  • 36.3k
1 vote
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Are differential equations beyond coordinates?

tl; dr: The coordinate-invariant expression of an ordinary differential equation is a vector field on a manifold. $\newcommand{\Reals}{\mathbf{R}}$In many circumstances, such as classical physics, ...
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1 vote
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Does a differential-geometry-isometry map a (sub)vector space to a (sub)vector space?

$\newcommand{\Reals}{\mathbf{R}}$Every Riemannian manifold can be isometrically embedded in a sphere; for a Sasaki metric the fibres map to bounded sets, which are therefore not affine. To flesh out ...
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1 vote
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Sobolev embedding implies lowerbound on the volumn of the ball

Let $r<R$. Then $\mathrm{Vol}_{g}(B_{x}(r)) \leq \mathrm{Vol}_{g}(B_{x}(R)) \leq (1/2C)^{n}$. Hence $\mathrm{Vol}_{g}(B_{x}(r)) \leq (1/2C)^{n}$, which is equivalent to $C \leq \mathrm{Vol}_{g}(B_{...
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  • 1,375
1 vote

Do Carmo Riemannian Geometry, definition 2.6

As an answer to your first question, $f$ is indeed real-valued. Here is a reference on the use of the terms "mapping" and "function" in differential geometry. Even though Do Carmo ...
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  • 95
1 vote
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Lee Smooth Manifolds Theorem 6.23's Jacobian Matrix

Well, the matrix of a linear operator is dependent on what basis you choose, so you have to identify the coordinates on the tangent bundle: Say $$v = a^i\frac{\partial}{\partial x^i }$$. Then $(x_1, \...
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  • 2,781
1 vote
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Isotropy subgroup of $SL(2,\mathbb{C})$ action.

We can also do this by using the extended complex plane for $\mathbb{P}^1$ and Mobius transformations $(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix})z=\frac{az+b}{cz+d}$. We are ...
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  • 14.6k
1 vote

Metric tensor for resulting manifold defined at coordinate $x$ of n-dimensional manifold $\mathcal{M}$

Let $ds^2$ be the Riemannian metric on $M$ (which is an $n$-dimensional manifold). Then for the map $F(x)=(x,f(x))$, the pull-back $F^*(ds^2 + dt^2)$ of the product metric $ds^2 + dt^2$ on $M\times {\...
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