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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16 views

$\varphi$-related vector fields when $\varphi$ is an inclusion

I'm reading Jeffrey Lee's Manifolds and Differential Geometry. He's talking about vector fields being $\varphi$-related to each other. He says If $S$ is a submanifold of $M$ and $X \in ...
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28 views

Short question about differential forms / exterior algebra

I am working towards understanding the wedge product of two vectors. Here is what I have so far: Let $V$ be an $n$-dimensional vector space over $\mathbb R$ (or $\mathbb C$, I'm not sure it makes ...
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28 views

Tangent bundle of an almost complex manifold

Let $(X,J)$ be an almost complex manifold with dimension $2n$. Then the tangent bundle $TX$ can trivially be made an $n$ dimensional complex vector space along each fibre. But how can I find a smooth ...
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30 views

Proof of the Beltrami theorem

I'm studying from a textbook and came across a theorem as the following, which it calls the Beltrami Theorem: Beltrami Theorem: A Riemannian metric $g$ is projectively flat if and only if it is ...
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43 views

How to interpret $a_i^k g_{kj}g^{jl}$? (Tensor index notation)

I have the equation $$a_i^k g_{kj}=-b_{ij},$$ where $a_i^k$ are given by: $a_i^k \cdot\vec r_k=\vec n_i$. Here, $g_{ij}$ and $b_{ij}$ are the matrices that determine the first and second fundamental ...
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22 views

When does a Lie bracket exist for a Frechet manifold?

Does a general Frechet manifold admit a Lie bracket? A bracket can certainly be constructed in certain cases, but my guess is that it is wrong to assume that one exists in general.
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24 views

Normal and tangent vectors to a curve

Assume we have a displacement field $u=u(x,z)=(u_x(x,z), u_z(x,z)))$ which takes the point $R=(x,z)$ to $r=R+u=(x+u_x(x,z), z+u_z(x,z))$. We want to find the tangent and normal to the curve which was ...
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29 views

Operators of order $k$ between Sobolev spaces

It may sound like tautology, but I have a problem in proving that differential operator of order $k$ is an operator of order $k$. What exactly I'm asking for? Let $M$ be a manifold and $E,F$ be two ...
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29 views

Étale morphisms definition?

Working over a commutative ring $R$, let $D= \left\{ d\in R : d^2 =0 \right\}$ A formally étale morphism $f:M\rightarrow N$ is one for which the square below is a pullback for every point ...
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35 views

Gaussian curvature of $S^3$

It is easy to see that Gauss curvature of $S^2$ is $1/R^2$. How can we find the Gaussian curvature of $S^3$? What about $S^n$ in general?
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31 views

$\exp_{x}$ is only $C^{1}$ at $y=0$.

According to the following image of book "Riemann-Finsler geometry" by chern & shen I would like to know which theorem of ODE theory is applied? Thanks.
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29 views

Necessary condition for the intersection of two submanifolds to be a submanifold

Let $X$ be an $n$ dimensional manifod. How could I prove that for arbitrary submanifolds $M,N$ of dimension $n-m,n-k$, if $\forall x\in M\cap N$ $dim( T_xM\cap T_xN)=n-m-k$ then $M\cap N$ is a ...
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26 views

Tightening a loop

Consider two $d$-dimensional convex polytopes $c_1, c_2$ that share a $(d-1)$-dimensional face $f$. Let $M$ be a surface ($2$-manifold) that intersects each of $c_1$ and $c_2$ in a $2$-ball. Suppose ...
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8 views

On the algebra of functions of an embedded manifold

We know that we can embed a manifold $\mathcal{M}$ of dimension $n$ in $\mathbb{R}^m$ with $m$ sufficiently high and specify the embedding using $n-m$ relations for the ambient coordinates. The ...
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42 views

Describe an atlas of smoothly related charts for the Special Orthogonal Group $SO(3)$

The Special Orthogonal Group $SO(3) =$ {${A \in M_3(\mathbb{R}) : A^TA = I, det(A) = 1}$} I have successfully shown that $SO(3)$ is a manifold, but I am having a difficult time explicitly finding a ...
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54 views

How to express elements of a first and second fundamental forms by their eigenvalues

Is there a way to express elements of a symmetric non-singular $2\times2$ matrix through its eigenvalues? Physics behind: I have an expression $f({\bf I},{\bf II})$ depending on the first and second ...
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40 views

vector space of manifold with/without boundary

I start learning Manifold. So I am wondering one question that "What is the difference between tangent vector space at a point of manifold with boundary and tangent vector space at a point of ...
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64 views

Prove topological space has countable basis

Given a topological subspace M of $\mathbb{R}^2 \times S^1$ defined by $(x,y,e^{i\theta})$ and two charts $(U,h)$, $(V,k)$ such that $H:\mathbb{R} \times (-\pi,\pi) \to M$ $H(x,\theta) = ...
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34 views

Counterexample of the fundamental theorem for hypersurfaces in the Euclidean space $\mathbb{R}^{n+1}$ in global sense

Are there any example of a closed Riemannian $n$-manifold $(M,g)$ and a symmetric bilinear form $A=h_{ij}dx^i\otimes dx^j\in\Gamma(T^\ast M\otimes T^\ast M)$ satisfying Gauss' equation ...
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31 views

$SO(3)$ and twisting the 2-sphere

I am currently reading some parts of "Rotating Relativistic Stars" by Friedman and Stergioulas and I have to say mathematics should NOT be taught by astrophysicists... Anyway, I've encountered the ...
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26 views

Smooth of $d(x_0,exp_x(v))$ and injectivity radius.

According to this question, the curvature can't control the injectivity radius. So, I don't know why support of $\varphi(v)$ need be small compared to maximum curvature? I think it should be compared ...
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31 views

Description of Frechet derivatives and the implicit function theorem

[QUESTION] Let $(S^n,\bar{g})$ be the unit sphere and $h$ be another Riemannian metric on $S^n$, $0<\alpha<1$. $M^{2+\alpha}(S^n):=\left\{F:S^n\stackrel{C^{2,\alpha}}\to S^n\right\}$. For ...
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10 views

Barred and unbarred systems

Say I have the set of linear equations $\bar x = 5x-2y$ and $\bar y= 3x+2y$ which define a linear transformation. The image of the point $(0,-1)$ is $(2,-2)$ Similarly $(2,1)$ has the image ...
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32 views

Difference between regular and smooth curve?

I have question about curves in space. What is the difference between regular curve and smooth curve?
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40 views

Helmholtz-Decomposition-weak Formulation in n-dimensional case

In the Wikipedia article about the Helmholtz-Decomposition it says in the section about Weak Formulation: For a slightly smoother vector field u ∈ H(curl, Ω), a similar decomposition holds: ...
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52 views

irrotational vector-field => Existence of scalar potential - for Sobolevfunctions

For $v\in C^1(\Omega,\mathbb{R}^n)$ the following is well-known: Let $\Omega \subset \mathbb{R}^n$ be simply connected. Then for every $v\in C^1(\Omega,\mathbb{R}^n)$ with $curl \;v = 0$ there exists ...
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49 views

De Rham cohomology ring of the Stiefel manifold in low dimensions

Let $V_k(\textbf{R}^n)$ be the Stiefel manifold : the $k$-frames in the $n$ dimensional real space. I'm trying to understand the de Rham cohomology ring for $k=2$ or $k=n-1$. I had good ideas for ...
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32 views

Symmetric Polynomials in Geometry

I'm interested in symmetric polynomials. Could you name some nice examples in Differential Geometry where they are clearly useful? I would also be interested in algebraic examples if connected with ...
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21 views

Seeing that the second fundamental form is the orthogonal component of the Laplacian

I have come across the statement a few times that, for a mapping $u:M\to N$ between a Riemannian manifold $(M,g)$ and a submanifold $N$ of Euclidean space $\mathbb{R}^n$, the part of the Laplacian of ...
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23 views

Evolute of Viviani's Curve

Can anyone explain to me why the evolute of Viviani's Curve is a single point. I'm afraid I don't understand why that is.
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50 views

About $T^6$ and $T^2 \times T^2 \times T^2$

I read often that wen can see the six torus like $T^2 \times T^2 \times T^2$. So, what is the difference between $T^6$ and $T^2 \times T^2 \times T^2$ ?
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local tangent approximation on manifold

Let $\mathcal{M}$ be a smooth $d$-dimensional manifold in $\mathbb{R}^n$ with "bounded curvature" (i.e. positive reach). $A(p)$ is small neighborhood around $p\in\mathcal{M}$ and $\tilde{A}(p)$ is its ...
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26 views

what will be the del operator

if the curve in $ x,y,z$ is parameterized as $(r(\gamma )\cos (\theta),r(\gamma ) \sin(\theta ), z(\gamma ))$. where the curve in $x,z$ is rotated about $z $ axis With $\gamma$ being the arc length. ...
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13 views

Basis vectors for “perturbed slicings” of a function, using SE(3)

Given a function $\Phi: \mathbb{R}^3\rightarrow\mathbb{R}$, we intruduce its planar cross section slices $\phi^{s}:\mathbb{R}^2\rightarrow\mathbb{R}$, using a rigid mapping $s \in SE(3)$ such that for ...
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19 views

calculation of mean curvature?

Converting a SDE from stratonovich to ito's form , Stratonovich form $$\partial X=P(X)\partial B$$ $$ P(X)=I-n(X)n(X)^T $$ Ito's conversion $$ dX=P(X)dB +\frac{1}{2}d(P(X))dB $$ $$ dX=P(X)dB ...
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24 views

Tangentbundle of the Tangentbundle

I´m just getting started in differential geometry and the following question came up to me: For any smooth manifold $\; M$ we can construct the so called tangentbundle $TM$. Since $TM$ is itself a ...
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33 views

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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29 views

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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148 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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21 views

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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28 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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22 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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15 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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60 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 ...