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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Is a stretched out torus still a $C^\infty$ manifold?

Suppose you have a torus and you carefully make a cylindrical cut down the center. Then you stretch out the outer half and glue together annular regions of the plane in the empty space. Now you have a ...
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30 views

Definition of strong tangent.

Let $\alpha:I\rightarrow \mathbb{R}^3$ a parametrized curve. What is the definition of strong (weak) tangent of $\alpha$ at the point $t_0$? Thanks!
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The level set of a smooth function

Let $f$ be a smooth function on a manifold $M$. Fix a point $p\in M$ and let $df\in T^\ast_pM$ be the differential of $f$ at $p$. I read that the subspace of $T_pM$ of vectors $X$ such that $df(X)=0$ ...
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Picard theorem on ODE, question with initial data

How one can prove, that the solutions depend smoothly on the initial data for Picard Theorem on ODE?
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Simple question on symmetric tensors 2

This question is related to this one Simple question on symmetric tensors. To prove that a vector field $Z$ is Killing, we use the identity $$0=(L_Zg)(X,Y)=g(X,\nabla_YZ)+g(\nabla_XZ,Y)\ \ \ \forall ...
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If a connected Lie group is divisible, is its exponential map surjective?

A group $G$ is divisible if for all $g \in G$ and $k \in \mathbb{Z}_+$ there is an element $h \in G$ such that $h^k = g$; we call such an $h$ a $k$th root of $g$. In an answer to a recent question ...
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Connected Sum: Actual Example

Suppose I have two tori as in the image: I have parameterizations of each torus and I want to form a nice $C^\infty$ connected sum. How do I do this? I know the theory, but not the practice. How do ...
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118 views

Covariant differential on p forms

On Peter Li's book Geometric Analysis on page 19 (http://www.im.ufrj.br/andrew/GR14-2/Lecture%20Notes%20on%20Geometric%20Analysis.pdf) I can't understand the following line $\begin{array}{l} ...
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set of almost complex structures on $\mathbb R^4$ as two disjoint spheres

The set of almost complex structures on $\mathbb R^{2n}$ is given by $$ M_n = \frac{GL(2n,\mathbb R)}{GL(n,\mathbb C)} = \mathcal C_+ \sqcup \mathcal C_-,$$ taking into account that $\det = \pm 1$ ...
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35 views

A converse proof that involves Torsion, curvature, and differentiation that equates to 0

I am having difficulty proving the converse in part B. I understand part A and have shown that t/k-t/k = 0. I found that n=-1/k, n'-bt= b'+tn = 0, so n' = bt and b'=-tn. However, I am unable to find ...
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30 views

Category of pointed manifolds

Let consider the following data: the family of pairs $(M,p)$ with $M$ a smooth manifold and $p \in M$ for every pair $(M,p)$ and $(N,q)$ as above a set $\hom[(M,p),(N,q)]$ whose elements are germs ...
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40 views

Difference between the Jacobian matrix and the metric tensor

I am just studying curvilinear coordinates and coordinate transformations. I have recently come across the metric tensor ($g_{ij}=\dfrac{\partial x}{\partial e_i}\dfrac{\partial x}{\partial ...
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18 views

Is the image under a homeomorphism of the cut locus $C_p$ a null-set?

Let $M$ be a complete Riemannian manifold with a point $p  \in M$ and let $U \subset T_pM$ be an open disk containing $0_p$ in the tangent space to $p$. By $C_p$ we denote the image of the boundary of ...
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41 views

Rotation Matrix to Quaternion(proper Orientation)

Given Data in the figure In this figure we have a unit vectors $ x,y,z$ as axis. Axis of rotation is $b$ and angle of rotation is $\phi$. $\phi$ is unknown and $b$ is given as $b= \frac{1}{2 ...
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53 views

A proof that involves Torsion, curvature, and differentiation that equates to 0

I know that I am supposed to write alpha = lambda*T + mu*N + v*B then differentiate then use the fact that {T,N,B} is a basis in R^3. I am just unsure how to write alpha as a combination using the ...
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24 views

Taylor development on manifolds and Manifolds of differentiable Mappings?

I'm trying to study the book "Manifolds of Differentiable Mappings" written by Michor and I came across the following: He considers two smooth manifolds $M$ and $N$ and define an equivalence relation ...
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24 views

connection on a vector bundle. horizontal spaces canonically isomorphic to horizontal spaces in the projection

Let $E$ be a vector bundle over $M$. A connection on a vector bundle $E$ is a smooth field of horizontal spaces $v \in E \mapsto H_v$. Where the projection is $\pi:E \rightarrow M$, $V_v$ is the ...
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86 views

A curve internally tangent to a sphere of radius $R$ has curvature at least $1/R$ at the point of tangency

Suppose $a$ is an arc length-parametrized space curve with the property that $\|a(s)\| \leq \|a(s_0)\| = R$ for all $s$ sufficiently close to $s_0$. Prove that $k(s_0) \geq 1/R$. So, I was going ...
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26 views

Quaternion Solution of the Rotation Equation

I am trying to make a connection between a 3-d vector ODE with a quaternion ODE and a possible solution in quaternion. In the following, a vector $v$ in $R^3$ is interpreted as the vector part of the ...
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48 views

Tractricoid as a pseudosphere (surface with constant negative curvature)

How to motivate/calculate/prove see that the tractricoid, i.e. a tractrix rotated about its asymptote, has a constant negative curvature? What are the hyperbolic lines on a tractricoid and how to see ...
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42 views

Relation between a function on $N$ sphere and a function on $(N-1)$-cell.

Let $S^N$ be a unit $N$-sphere. Let $f:S^N\to\mathbb{R}$ be a function. Let $\bf{\Sigma}$ be a unit $(N-1)$-cell, consider the function $g:S^N\to\bf{\Sigma}$ such that, for any $\hat{a}\in S^N$, ...
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41 views

Do Taylor series of analytic vector field depend on metric tensor?

Let $\mathcal{M}$ be $n$-dimensional Riemannian manifold. And let $F$ be an analytic vector field on it. By definition this means that $$f^i(x(q)) = f^i \vert_{x(p)} + \frac{\partial f^{i}}{\partial ...
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Geodesic deviation on a unit sphere

No response to this on Physics Stack Exchange, so I'm hoping for better luck here. My question is, can anyone tell me where I'm going wrong trying to use the equation of geodesic ...
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Non-vanishing differential forms and flatness

Let $M$ be a differentiable manifold of dimension $n$. If the tangent bundle is trivial, then the cotangent bundle is trivial, and so are its exterior powers. In other words, on a parallelizable ...
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33 views

Simple question on symmetric tensors

This question seems to be silly, but i am really confused. Suppose we have a symmetric $2$-tensor $\omega$, I want to prove $$\omega(X,Y)=0\ \ \ \forall \ \ \ X,Y \iff \omega(X,X)=0 \ \ \forall \ \ \ ...
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47 views

Problem with a Do Carmo problem

I'm trying to solve some Do Carmo problems from his book Differential geometry of curves and surfaces. In section 1-3 prob.5.c., we have the curve: $\alpha:(-1,\infty)\rightarrow\Bbb R^2$ given by: ...
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cohomology ring of grassmannian

Let $G_k(\mathbb{R}^n)$ be the grassmannian consisting of all $k$-subspaces in $\mathbb{R}^n$. How to compute the cohomology ring $$H^*(G_k(\mathbb{R}^n);\mathbb{Z})$$ and what is the result?
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definition of critical point defined in terms of differential map

I am having a problem understanding the definition of a critical point in do carmo's Differential Geometry of Curves and Surfaces. He notes in page 58 that a point $p \in U $ is a critical point of ...
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use Noether normalization theorem to integrate differential forms over singular subvarieties

Let $X \subset \mathbb C^n$ be an analytic subset. I would like to show that locally around any point $x \in X$ the regular part $X_\text{reg}$ has finite volume, perhaps using the theorem below. ...
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45 views

Metric tensor and einstein's notations .

Here , $\Omega \subset \mathbb R^n$. Can someone explain it to me what $F$ is ? I also don't understand how we can get the unit outer normal in the second para . Please kindly help me to understand ...
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249 views

Is the following solution to the isoperimetric problem correct?

I came up with a solution over here: http://keplerlounge.com/ but I am not entirely sure that it's free from error as much as I have checked it many times. If people can offer constructive feedback ...
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34 views

Radial & perimeter growth and Gauss curvature fall

A flat circular patch radius $r$ , initial perimeter $2 \pi r$ and initial patch area $ \pi r^2$ grows (dilates) or shrinks, non-isometrically. Shrinkage is associated with radial and perimeter ...
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about moments of a uniform distribution on a high-dimensional ball

I need to understand how the following integrals depend on the dimension $d$; the result should be about a (negative) power of $d$. Let $\mathbb{B}^d$ be the $d$-dimensional ball of radius $1$, ...
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Identity concerning push forward of two vector fields

How would you prove the identity $\displaystyle \frac{\partial}{\partial s}\Psi_{s^*} \mathbb{X} = (-1)L_{\mathbb{Y}}\Psi_{s^*}\mathbb{X}$ where $\Psi_{s}$ is the flow of $\mathbb{Y}$ and ...
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1answer
43 views

tangent bundle and normal bundle

I have a problem about tangent bundle. It is known that the tangent bundle of most manifolds is not trivial: for example, the tangent bundle for $S^2$ is not $S^2\times \mathbb{R}^2$. However, for a ...
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Characterization of the exterior derivative $d$

In the paper Natural Operations on Differential Forms, the author R. Palais shows that the exterior derivative $d$ is characterized as the unique "natural" linear map from $\Phi^p$ to $\Phi^{p+1}$ ...
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How do I map Df(w) to it's [lie] group/algebra representation?

E.G. For $p,w\in(\mathbb{R}^3,+,\times_\vartheta)$ with $(\mathbb{R}^3,+)$ a vector space and with $p=(r,s,t)$, $w=(x,y,z)$ where we have $p\times_\vartheta ...
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17 views

Maximize first co-ordinate on general ellipsoid

I have an ellipsoid of the form x^TAx=k , where A is 3x3, positive definite and symmetric. I need to find maximum x(1) over the ellipsoid. Can I maximize x(1)^2 by taking partial derivative of the ...
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64 views

show $\omega$ is exact form

Let X be the region $\mathbb{R^3}-(0,0,0)$ and f(x,y,z) is $C^\infty$ function on X. Also $\omega$ is 1-form $f(x,y,z)(xdx+ydy+zdz)$. if $f$ can be expressed in the form $f(x,y,z)=h(r)$ when ...
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Finding an isometry that maps one circle to another.

I have a problem goes as follows: Consider the unit speed curve $$\boldsymbol{r}(s)=\left(\frac{4}{5}\cos(s),1-\sin(s),-\frac{3}{5}\cos(s)\right).$$ Find an isometry $f$ such that ...
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48 views

A Problem from Docarmo's Differential Geometry

The following is a (may be simple) problem from Docarmo's Differential Geometry. Let $\alpha\colon (a,b)\rightarrow \mathbb{R}^3$ be a parametrized curve which do not pass through origin. If ...
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Is an configuration space contractible?

Let $\mathbb{R}^\infty$ be the union $\cup \mathbb{R}^n$(with inclusions). Let $F(\mathbb{R}^\infty,k)$ denote the $k$-th configuration space of $\mathbb{R}^\infty$, i.e., the subspace of ...
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Exponential of a power of the differential operator

In relation to this question: Exponential of a polynomial of the differential operator Is there an expression for $\exp(aD^n)f(x)$ similar to $\exp(aD)f(x)=f(x+a)$?
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What does this operator $\odot$ mean

I read this about the second fundamental form in Wikipedia and I’ve no idea what does $\odot$ mean? Does anybody know? $$II=-dN\cdot dP=\omega^3_1\odot\omega^1+\omega^3_2\odot\omega^2$$
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defining smooth functions on manifolds *without* smooth chart transitions

Let $M$ be a topological manifold, covered by an atlas of charts ${(U,\phi_U)}$ (which are homeomorphisms into Euclidean space), and let $p\in M$. Say a function $f:M\to\mathbb{R}$ is smooth at $p$ if ...
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Differential Geometry Intuition Question

Apologies if I get the notation wrong. Still learning this stuff. Suppose I have a 2 dimensional Riemannian manifold $\mathcal{M}$ that is covered by a single chart: $\phi: \mathcal{M} \rightarrow ...
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48 views

Hessian is proportional to the metric everywhere

Let $(\Omega^{n+1},g)$ be a compact Riemannian manifold with smooth boundary. Let $f\in C^{\infty}(\bar{\Omega})$ satisfies $\operatorname{Hess}f=\frac{1}{n+1}g.$ Suppose the minimum of $f$ occures at ...
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38 views

defining smooth functions on smooth manifolds

The standard approach to defining smooth functions $f:M\to\mathbb{R}$ on a topological manifold $M$ equipped with a smooth structure (i.e., a maximal smooth atlas) $\mathcal{A}$ is the following. Say ...
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map between classifying spaces induced by group homomorphism

Let $\Sigma_n$ be the permutation group of order $n$. Then the regular representation of $\Sigma_n$ gives an injective homomorphism $f:\Sigma_n\to O(n)$. Why $f$ induces a map between their ...
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44 views

Use contraction mapping theorem to prove

Help! I am taking a math course, and I just can't figure out this proof: Let $\alpha,\beta\in R^n$, $a\in R$, and $A$ be an $n\times n$ nonsingular matrix. Use contraction mapping theorem to prove ...