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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Why a sphere cannot have a Lorentzian Metric?

I was listening to a Lecture and the lecturer said that a sphere cannot have a Lorentzian Metric. Is that accurate? If so, why?
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Explaining the disinvolutivity of a distribution in light of an integral submanifold

Here is the problem 19-5 of Lee's introduction to smooth manifolds. Let $D$ be the distribution of $\mathbb{R}^3$ spanned by \begin{align*} X&=\frac{\partial}{\partial ...
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149 views

Smoothness of solutions of the curve shortening flow given bounded curvature

I've been looking at the Lemma 1.5 of The Heat Equation Shrinks Embedded Plane Curves to Round Points (here), where Matthew Grayson proved that Suppose that $\kappa$ is bounded for $t\in[0,t_0)$. ...
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Map of constant rank

Let $f_1, \dots, f_m \colon M \to \mathbb{R}$ be smooth functions on a manifold $M$, such that $F := (f_1, \dots , f_m) \colon M \to \mathbb{R}^m$ has constant rank $k <m $ on M. I'm trying to ...
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29 views

Simple exercise in differential geometry

Problem: Prove the identity $V=\sum V[x_i]U_i$, where $x_1, x_2, x_3$ are the natural coordinate functions. (Hint: evaluate $V=\sum v_i U_i $ on $x_j$) Elementary differential geometry written by ...
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23 views

Topologies on Spaces of k-rectifiable sets, and $C^k$ convergence

I've been thinking about topologies on the spaces of k-rectifiable sets (Hausdorff metric topology, varifold topology, etc) in $\mathbb{R}^n$, and I'm wondering if there are hypotheses under which ...
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17 views

When does contractible space of almost complex structures taming a given symplectic form $\omega$ contain an integrable compatible one?

Given a symplectic form $\omega$ on a compact symplectic manifold $X$, we know there is a contractible homotopy class $\mathcal{J}_{\omega}$ of almost complex structures that tame $\omega$. A subset ...
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16 views

Bounding the distance between two points based on their 1st and 2nd derivatives

Is there a known way to place a bound the distance between two points - here loosley speaking two minima - knowing their first and second derivatives at those specific points? Say I have two convex ...
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38 views

Let $\mathcal{L} =\{x+iy:x,y\in \mathbb{R},F(x,y)=0 \}$ is algebraic curve. How Can we prove, $\mathcal{L}$ is a piecewise $C^\infty$curve

Suppose $F(x,y)$ is polynomial . $\mathcal{L} =\{x+iy:x,y\in \mathbb{R},F(x,y)=0 \}$ . We know that ${\cal L}$ is algebraic curve. How Can we prove, $\mathcal{L}$ is a piecewise $C^\infty$curve, ...
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55 views

Prove that the concept of a vertex does not depend upon the parametrization

Any hints on how to start the proof? The curvature of a regular curve $\beta(t)$ (not necessarily unit speed) can be written as: $\kappa$ = $\frac{|\beta'(t) \times \beta''(t)|}{|\beta'(t)|}$. Then ...
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13 views

A-paths on Lie algebroids?

In the paper integrability of Lie brackets M. Crainic and L. R. Fernandez define the notion of A-path as follows. Definition. Let $A\stackrel{\pi}{\longrightarrow} M$ be a Lie algebroid. An A-path is ...
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61 views

Vector Laplacian operator in orthogonal curvilinear coordinates

I'm looking for a simple expression for the vector Laplacian $\nabla^2\mathbf{A}$ in orthogonal curvilinear coordinates. Actually, I don't require the whole thing, just the part of ...
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56 views

Volume-preserving Diffeomorphism

Anybody knows the conditions that a function $\gamma(x,t)$ with domain $(x,t)\in[0,1]\times[0,+\infty)$ and satisfying $\gamma_x(x,t)>0$ for all $x\in[0,1]$ and $t\geq 0$ must satisfy in order for ...
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38 views

Proving the transition map of a smooth surface is smooth

First of all, this is not a homework problem, so please do give whatever advice/criticism you can give. I'm reading Pressley's Elementary Differential Geometry and he gives the theorem that ...
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45 views

Explicitly understanding the implicit function theorem

Suppose I have a curve $f$ in $\mathbb{R}^2$, the implicit function theorem guarantees the existence of a smooth local inverse of this function $f$. Question: My question is is there a way to ...
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34 views

Leibniz Integral Rule for the indefinite integral

Can I use the Leibniz Integral Rule for the following indefinite integral? $$\frac{\partial^{2}}{\partial s \partial t}[\int |f(x)+ t g(x) + s h(x)|dx]^{2}|_{s=t=0}$$ where $f,g,h:[0,1]\to ...
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24 views

Differentiable stacks and morita morphism

I heard that if $[X_0/G_0]$ and $[X_1/G_1]$ are differentiable stacks, then any morphism between them is naturally equivalent to $$(G_0 \rightrightarrows X_0) \xleftarrow{\simeq} (G_2 ...
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37 views

Side Roll “Catenoid” pair

Apologies in advance for the very poor quality of images. Longitudinal stretching a catenoid of revolution results in well-known isometric helicoidal deformation. Lateral stretch or spread-out ...
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25 views

reversal of constant rank theorem

Assuming $M$ is a manifold of dimension $\dim M = n+k$ and $F \colon M \to \mathbb{R}^n$ is a smooth function, such that the levelsets $F^{-1}({c})$ are submanifolds of dimension $k$. Can I therefore ...
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76 views

Advanced definition of derivative.

In the paper On proof and progress in mathematics, W. P. Thurston gives the following interpretation of the derivative. ...one person’s clear mental image is another person’s intimidation: ...
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36 views

Are there any (differential) geometries of interest which cannot be formulated as Cartan geometries?

Cartan geometry is a generalization of Klein's Erlangen program. For every homogeneous model space, we have a corresponding type of Cartan geometry obtained by "rolling without sliding" the model ...
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53 views

Integral of the square root of a trigonmetric function

Despite my best attempts, I have been unable to evaluate the following integral: $$ \int_s^t\sqrt{9+(2+\cos3u)^2}\,du. $$ This integral showed up during an investigation of torus knots. It represents ...
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35 views

Eigenfunctions of the Dirichlet Laplacian in balls

I am trying to find out about the Dirichlet eigenvalues and eigenfunctions of the Laplacian on $B(0, 1) \subset \mathbb{R}^n$. As pointed out in this MSE post, one needs to use polar coordinates, ...
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Does this container exist?

EDIT: Note that the object I'm seeking needn't have anything to do with water or actual containers; those are just used to convey the idea. I'm trying to find a container that, when turned with ...
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31 views

Riemann curvature tensor for geometry surfaces on $\mathbb{R}^3$

Let $M\subseteq \mathbb{R}^3$ be a regular surface and $p\in M$. The Riemann curvature tensor is defined by: $$\begin{array}{rcll} R_p:&T_pM\times T_pM\times T_pM&\longrightarrow &T_pM\\ ...
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33 views

moment of inertia and parallel axis theorem

A lamina with density $\delta \left ( x,y \right ) = x^{2}$ has the shape of the disk $\left \{ \left ( x,y \right )|x^{2}+y^{2}\leq 4 \right \}$. Find the moment of inertia of the lamina about the ...
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Exponential map and $\lim_{v,w \to 0} \frac{d(\exp(v),\exp(w))}{\|w-v\|}$

Let $v,w \in T_{p}M$. Prove that $$\lim_{v,w \to 0} \frac{d(\exp(v),\exp(w))}{\|w-v\|}=1$$ I completely don't know how to start. Thanks for any hint. It is an exercise to lecture based on ...
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35 views

Differentiable, Parametrized Curve for Trace $y = |x|$

my professor gave practice problems for an upcoming midterm, and one is to find a differentiable parametrized curve with trace $y = |x|$ with $-1 \leq x \leq 1$. My thinking is that I should find a ...
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24 views

Every non-constant closed curve has positive period

I want to show that every non-constant closed curve has positive period, but i'm not really sure how to do this. A smooth curve $r(t): \mathbb{R} \to \mathbb{R}^n$ is $T$-periodic if $r(t+T)= r(t)$ ...
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26 views

Upper bound on hessian

Given a smooth Riemannian manifold $(\mathcal{M},g)$ and $f \in C^{\infty}(\mathcal{M})$ let $r(x)= d(x,x_0)$ where $d$ is the distance function wrt $g$ and $x_0$ is some point on the manifold. If we ...
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19 views

Regarding the axis of screw motion for a space curve.

The axis of the accompanying screw motion of a curve $c(s)$ at any point $c(s_0)$ is the line in the direction of the Darboux vector $\tau(s_0) T(s_0) + \kappa(s_0)B(s_0),$ through the point $$P(s_0) ...
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30 views

Extending automorphisms on surfaces

Assume we are in the complex setting. Let $X$ be a surface, $C$ a curve on $X$. Say $X-C$ is isomorphic to some $X'-C'$ whith $X'$ a surface and $C'$ a curve on $X'$. If it helps we may assume that ...
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30 views

Why is the backwards curve needed for the inverse of a parallel transport?

Dearh math.stackeschange-community, I'm at a loss with the following problem: Let $I=(a,b)$, then the inverse of the parallel transport $P_{s,t}^\gamma$ from $s \in I$ to $t \in I$ along the curve ...
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21 views

Showing that $(\mathbb{R}, \mathscr{F})$ and $(\mathbb{R},\mathscr{F_1})$ are diffeomorphic but $\mathscr{F}\neq \mathscr{F_1}$

Background $M$ is locally Euclidean with dimension $d$ if $M$ is hausdorff and every point in $M$ has a neighborhood homeomorphic to $\mathbb{R{^d}}$. If $U\subset M$ is open and connected and $\phi$ ...
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29 views

Hessian of the Stereographic projection

Consider the stereographic projection from the sphere $S^n$ onto $\mathbb{R}^n$, and take the usual local spherical (polar) coordinates $\omega_1,..,\omega_n$ on $S^n$ (coming from its embedding in ...
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41 views

Killing vector field for Witten complex?

I am reading a classical paper by Atiyah Bott "The moment map and equivariant cohomology". In paragraph "Relation with Witten complex" (at the very beginning of this paragraph) they claims that ...
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47 views

Hodge theory in general

I know a bit of Hodge theory, and I know that there is an analogue in the symplectic case, where instead of inducing the $\star$-product using the metric we use the symplectic form. Is in true in ...
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21 views

when can a surface conformally equivalent to the sphere be isometrically immersed?

Given a scalar function $s:S^2\to \mathbb{R}$, and the induced Euclidean metric $g$ on $S^2$, when can the sphere, equipped with metric $e^sg$, be isometrically immersed in $\mathbb{R}^3$? Is there a ...
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28 views

When does a non singular integrable differential one-form define a regular foliation?

Let $\mathcal{M}$ be a smooth manifold of dimension $m<+\infty$. Let $\theta$ be a nowhere vanishing (non-singular) differential one-form on $\mathcal{M}$ such that $\theta\wedge d\theta=0$. ...
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29 views

The first eigenvalue for Dirichlet boundary condition positive?

Let $M$ be a compact, n dimensional Riemannian manifold with boundary. Then we know that $W^{1,2}(M)=W^{1,2}_0(M)$, the latter is the completion of $C_0^{\infty}(M)$ function w.r.t $W^{1,2}(M)$-norm. ...
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A question about notation in derivatives inn $\partial_{\bar{A}}L^I$

In this paper http://arxiv.org/abs/1210.2332, when the authors say eq (2.17): $$\partial_{\mu}L^I=\partial_{\bar{A}}L^I\partial_{\mu}\bar{z}^A+\partial_AL\partial_{\mu}z^A$$, does they mean by ...
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35 views

Proving smooth map between smooth manifolds is constant based on push forward being zero

I have just me this problem in my class on smooth manifolds from Lee's introduction to smooth manifolds, from the chapter on the tangent bundle stating the following: Let M, N be smooth manifolds, ...
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45 views

normal bundles are stably isotopic

I am searching for a reference about the following theorem: Let $M \xrightarrow{i_{0}} \mathbb{R}^{k},\,\,$ $M \xrightarrow{i_{1}} \mathbb{R}^{l}$ be two embeddings of the manifold $M$. We know by ...
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24 views

tangent vectors

Let $S$ be any regular surface in $R^3$ and let $p \in S$ be any point. From Classical differential Geometry I Know that the tangent space of $S$ at p is a subspace of $R^3$. If I see the surface ...
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40 views

Checking a proof involving flows

I am going through the proof of theorem 2.12 of the book Lectures on the geometry of Poisson manifolds by I. Vaisman. It's just a bit of differential geometry, but as I don't use these methods very ...
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18 views

What is the most accessible reference on wall-crossing?

I am looking for a nice and easy to read reference on wall-crossing (in the context of Donaldson theory). Is there some accessible reference you have to suggest? I am interested in studying Donaldson ...
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17 views

Liegroup acting transitively => connected component acts locally transitive?

Suppose $G$ is a Liegroup acting transitively on a manifold $M$. Does that already imply, that the connected component $G^\circ$ of $G$ acts transitively on the connected components of $M$? I have a ...
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53 views

Well definedness of Lie bracket (coordinates approach)

For practice I wanted to define Lie bracet in terms of coordinates and show that this definition is independet of the choice of coordinates. So I started with the definition ...
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38 views

What does it allow to see Differential Geometry from an abstract viewpoint?

I am currently learning Differential Geometry from the "categorical" point of view. We use sheaves, ringed spaces, group objects,... to define smooth manifolds, vector bundles,... My previous course ...
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18 views

Curves with distance between them growing locally as $o(d^k)$

Context: I'm searching for some standard definitions related to order of contact between curves (and smooth manifolds in general). My research has taken me to the concept of jets. Simply speaking, a ...