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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Tangent space to noncompact Stiefel manifold

The noncompact Stiefel manifold is the set of $\mathbb{R}^{n \times p}$ matrices ($p \leq n$) that have rank $p$ (full rank). Based on my readings of ...
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27 views

Pushforward injective

Let $f : M \rightarrow N$ be a smooth surjective map between smooth manifolds. Now, consider a 2-form $\omega$ on $T_pN$. Does it now follow that the pullback satisfies? $f^* d \omega =0 \Rightarrow ...
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22 views

Deriving a curvature tensor for a special connection

May be $e(x)$ a frame vector at the point $x \in M$ for a manifold $M$. Given the following connection equation: $d_Xe+e_{|X_+}-e_{|X_-}= d_Xe + J_Xe = 0$. Here, $d_X$ denotes the directional ...
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38 views

Is there some relationship between algebraic curves and partial differential equations that goes beyond classifying different PDE's

I ask primarily because despite not having taken that many math classes (up to two semesters of a PDE class in college), it would be very interesting if maybe we could gain intuition regarding ...
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2answers
119 views

Isn't there a better way to put the canonical smooth structure on the $n$-sphere?

My differential geometry notes put a smooth structure on the $n$-sphere (denoted $S_n$) as follows. Firstly, $S_n$ is taken as a subset of $\mathbb{R}^{n+1}.$ Secondly, we define $U_0 = S_n \setminus ...
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Exterior derivative of a form and $d(d\omega)=0$?

We know that in differential geometry, $d^2\omega=0$, where $\omega $ is a form and $d$ is the exterior derivative. However if this form happens to be the exterior derivative of another form ...
2
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2answers
80 views

A “parallel manifold” is always orientable

I want to solve the following problem from Spivak's Calculus on Manifolds: Let $M$ be an $(n-1)$ dimensional manifold in $\mathbb{R}^n$. Let $M(\varepsilon)$ be the set of end points of normal ...
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1answer
26 views

If $M\subset \mathbb{R}^d$ is a manifold of dimension $m$ and $U\subset \mathbb{R}^d$ is open, then $M\cap U$ is not a open.

I'm reading notes about M-estimators, and have within these notes been briefly introduced to manifolds, as a way to create what the author call "smooth hypothesis" for statistical models. A basic ...
2
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28 views

Restriction of a Smooth chart is Again a Smooth Chart

Let $M$ be a smooth manifold with $\mathscr A=\{(U_\alpha,\varphi_\alpha)\}_{\alpha\in J}$ being its smooth structure. Let $V$ be open in $M$ and $(U,\varphi)\in \mathscr A$. Define $\psi:U\cap V\to ...
2
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29 views

Question about vectors in the Grassmannian in this example

Consider $f: \mathbb R^2 \to \mathbb R^3$ defined by $(t,s) \mapsto (t^2 + 2s, t^3 + 3ts, t^4 + 4t^2 s)$. Let $Gr$ denote the Grassmannian and let $Gr(2, T\mathbb R^3) = \bigcup_{x \in T\mathbb R^3} ...
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70 views

Integration by parts formula on unbounded manifold

Let $M$ be a closed Riemannian manifold and set $X = M \times [0,\infty)$ with the trivial product metric induced. If $u$ and $v$ are functions defined on $X$, how do I know that the formula $$\int_X ...
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1answer
19 views

Compute $(df)_a$ in chart $\varphi_1:U=\{(x,y,z)\in\mathbb{R}^3:x\neq0\}\rightarrow\varphi_1(U)$

Suppose that for a submanifold $H$ of $\mathbb{R}^3$ we have two charts $$\varphi_1:U=\{(x,y,z)\in\mathbb{R}^3:x\neq0\}\rightarrow\varphi_1(U)$$ ...
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What is the flaw in my thinking for the graph of this function?

Consider the map $$ f: \mathbb R^2 \to \mathbb R, (x,y) \mapsto x^3 + y^3 + xy$$ This defines a surface in $\mathbb R^3$. Let's consider some level set $f(x,y) = c$: (see here page 67) I think ...
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31 views

Incorrect statement in a proof of the transversality theorem?

I'm reading through Morris Hirsch's book on differential topology, and he makes the following offhand statement. Suppose k is a compact subset of a manifold U, and V is a vector subspace of R^n. If a ...
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20 views

Any unit speed reparametrization $\beta=\alpha(h)$ of $\alpha$ is reparametrized by an arc length function.

These are definitions given from Barret O'neill's Elementary Differential Geometry. Definition $1$. A curve in $\mathbb{R}^3$ is a differentiable function $\alpha: I\to \mathbb{R}^3$ from an open ...
3
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1answer
36 views

Mean Curvature Flow

Recently I am reading the mean curvature flow from the lecture notes of Carlo Mantegazza where I found that Under mean curvature flow given by$$\begin{cases}{\partial\over \partial ...
2
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1answer
26 views

Multidimensional variant of the fundamental Lemma of the Calculus of Variations

I wonder, if the following is true: Let $(M,g)$ be a compact Riemannian manifold and $f \in \mathcal{C}^{\infty}(M)$ be a smooth function. Then $f$ is constant if and only if for all $u \in ...
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1answer
13 views

Asymmetry of definition of regular value and critical value

Let $f: U \subseteq \mathbb R^n \to \mathbb R^m$ be a smooth map. We say $y \in \mathbb R^m$ is a regular value of $f$ if and only if all points in the set $f^{-1}(y)$ are regular. (see e.g. the ...
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45 views

Interior Derivative and Contraction: Kobayashi and Nomizu.

In Kobayashi & Nomizu, the interior derivative of an r-form is defined as $\iota_X \omega = C(X \otimes \omega)$, where $C$ is the contraction associated with the pair $(1,1)$ and $\omega$ is ...
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4answers
209 views

How many points can I prescribe for a diffeomorphism of the plane?

I was trying to find out how to construct a $\mathcal C^\infty$ curve that joins two arbitrary line segments. My idea was to use bump functions and the likes, but for that I had to make the line ...
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1answer
24 views

Congruence of two curves with an arbitrary speed?

I'm studying the book "elementary differential geometry" by o'neil. There is a collorary which states that if two curves a(t), b(t) which is defined in the same real line interval has the same speed, ...
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Show that the section $g(x_1,x_2,x_3)=x_1^2dx_1^2+dx_2^2+dx_3^2$ defines a Riemannian metric on $\mathbb{R}^3 - \{x_1=0\}$

Show that the section $g$ of $T^*\mathbb{R}^3 \otimes T^*\mathbb{R}^3$ defined by $g(x_1,x_2,x_3)=x_1^2dx_1^2+dx_2^2+dx_3^2$ defines a Riemannian metric on $\mathbb{R}^3 - \{x_1=0\}$ and compute ...
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46 views

Characterization of Semidirect Products: Too strong assumptions?

In John Lee's book Introduction to Smooth Manifolds, there is the following Theorem. Theorem 7.35 (Characterization of Semidirect Products). Suppose $G$ is a Lie group, and $N,H\subseteq G$ are ...
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9 views

How do I lay a time-varying sheet on a time-varying wire?

I ask this question because I think it will solve another question I posted on a PDE with a time-dependent boundary value. I'll pose the general question, then solve it in the case of $1$ dimension. ...
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1answer
43 views

Is this a sufficient condition for differentiability

Consider a function $f:\mathbb{R}^n\rightarrow \mathbb{R}$. Suppose that, for all $c\in \mathbb{R}$, every vector in $f^{-1}(c)$ is supported by a unique hyperplane to $f^{-1}(c)$. Is $f$ ...
2
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1answer
50 views

Properly discontinuous action on manifold

I am actually not familiar with topology, but since we had a short outlook on these things in our differential geometry lecture today, I would appreciate some general remarks: Let $M$ be a smooth ...
6
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2answers
517 views

Lie bracket is a connection?

In Road to Reality, section 14.6 on Lie derivative Penrose writes: Now $\epsilon^2 [j,h]$ corresponds to an $O(\epsilon^2)$ gap in the ‘parallelogram’ whose initial sides are $e_j$ and $e_h$ at ...
2
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1answer
35 views

Find an atlas for $M=\{(x,y,z)\in\mathbb{R}^3:x^2+y^2=1+z^2\}$

Find an atlas for $M=\{(x,y,z)\in\mathbb{R}^3:x^2+y^2=1+z^2\}$ It is easy for me to check that $M$ is a submanifold of $\mathbb{R}^3$ of dimension $2$ using the following theorem: Let $F:U ...
3
votes
1answer
59 views

Coordinate systems on manifolds

I am fairly new to differential geometry and something I can't get my head around is, if an $n$-dimensional manifold is locally homeomorphic to $\mathbb{R}^{n}$, i.e. Euclidean space, then isn't it ...
3
votes
2answers
41 views

Notion of curvature for a volume embedded in $R^3$

This question might sound slightly vague, but please bear with me. If I have an orientable, closed, sufficiently smooth surface in $R^3$, I can define its principal curvatures, mean curvature as ...
4
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36 views

Twisted geodesics on circular torus

I am attempting to make a Mechanics of materials approach to describe non-linear deformations of thin lines/wires on a torus. A mathematical modeling of its probable geometry is required at start of ...
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116 views

What is a moduli space for a differential geometer?

A moduli space is a set that parametrizes objects with a fixed property and that is endowed with a particular structure. This should be an intuitive and general definition of what a moduli space is. ...
2
votes
1answer
20 views

a curve does not need to be injective?

In diff. Geometry, curve is a differentiable mapping from an open interval to 3 dimensional euclidean space. Doesn't it need to be injective? If it is not, then there might be a two different tangent ...
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1answer
45 views

Finding incomplete geodesics

I have a problem with the notion of incomplete geodesics. Can someone give me a minimal example for such a geodesic? In particular, I am trying to solve the following exercise: Consider the upper ...
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1answer
36 views

Regular values on boundary of smooth manifold

Let $X$ be a smooth compact manifold without boundary and $Y$ be a smooth compact manifold with boundary $\delta Y$ where $\dim X = \dim Y$. Suppose $ f: X \rightarrow Y $ is smooth. As shown in ...
4
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1answer
28 views

Calculate the length of $\gamma(t)=(t,t), t \in [-1,-\frac{1}{2}]$ with the metric $g=\frac{dx^2+dy^2}{y^2}$ and compare with euclidean metric

Consider the metric $g=\frac{dx^2+dy^2}{y^2}$ on $\mathbb{R}_+^2=\{(x,y) \in \mathbb{R}^2 : y>0\}$. Calculate the length of the curve $\gamma(t)=(t,t), t \in [-1,-\frac{1}{2}]$ and compare ...
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0answers
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Constructing an immersion of a curve with certain properties

Consider the map $\gamma : \mathbb R \to \mathbb R^2$ given by $(t^2, t^3)$. Let $Gr(1,T\mathbb R^2) := \bigcup_{x \in \mathbb R^2} Gr(1,T_x \mathbb R^2)$ where $Gr(1,T_x \mathbb R^2)$ is the ...
3
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0answers
31 views

Minimum Knowledges to precisely calculate PDEs (integral equations)

Basically all I want to do is to calculate (or prove) precisely equations such as $$ \frac{1}{n\alpha(n)r^{n-1}}\int_{\partial B(x,r)} u(y) dS(y) = \frac{1}{n\alpha(n)} \int_{\partial B(x,r)} ...
1
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1answer
56 views

Curvature and Circumference of Circle

Theorem Let $\gamma\colon [a,b]\rightarrow \mathbb{R}^2$ be a unit speed simple closed curve, with $\gamma'(a)=\gamma'(b)$ and $N$ is the inward-pointing normal. Then $$ \int_{a}^b ...
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1answer
33 views

Can someone explain whether $C^{\infty}(M)$ is an algebra or a commutative ring?

One forms $\Omega^1(M)$ is a module over $C^{\infty}(M)$, therefore does that make $C^{\infty}(M)$ an algebra or a commutative ring?
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1answer
43 views

Find an atlas for $H=\{(x_1,x_2,x_3,x_4) \in \mathbb{R}^4 : x_1+x_2^2=x_3^2+x_4=1\}$

Find an atlas for $H=\{(x_1,x_2,x_3,x_4) \in \mathbb{R}^4 : x_1+x_2^2=x_3^2+x_4=1\}$ Let $F:\mathbb{R}^4 \rightarrow \mathbb{R}^2$ s.t. $(x_1,x_2,x_3,x_4) \mapsto (x_1+x_2^2-1,x_3^2+x_4-1)$. ...
8
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1answer
39 views

Embedding covers of manifolds

I am considering $k$-fold covers of smooth manifolds (with smooth covering maps). Let $f:M^m\to N^m$ be a smooth finite covering map. -- The following implication is not true: $M$ can be embedded ...
2
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1answer
46 views

$C^\infty$ bump function to smoothen a corner

Let $\beta(t)$ be a smoothener at $t=0,$ e.g. $\beta(t)=e^{-1/t},$ for $t\in \mathbb{R}^+.$ Let's say that this is a horizontal smoothener, as it flattens at $\beta(0^+)$. Now I want another function ...
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0answers
16 views

Showing tractrix revolution is isometric to hyperbolic plane

I have a tractrix defined (in the xy plane) by the fact that the line segment formed by a tangent to the curve meeting the x axis has length 1. Given that the tangent meets the x axis at an angle ...
4
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1answer
46 views

Coordinates on the sphere not global?

I'm reading a book on differential geometry and some part of the introduction I do not understand but I'm curious to understand it. Maybe someone can try to explain those parts to me. "Each point on ...
4
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0answers
36 views

Manifolds with 'bad metrics' (reference request)

While studying some differential geometry, a thought crossed my mind that I am sure has been considered before, but I cannot find a reference for it. What can be said about spaces for which the ...
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1answer
37 views

Find the curvature

If a point moves along a curve so that the velocity and acceletation vectors have constant lenght, how to proove that the curvature is also a constant?
2
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1answer
45 views

Existence of maximal tori in infinite-dimensional Lie groups

Reading about Lie groups and maximal tori I came up with a lemma that states that any Lie group $G$ has maximal tori. The proof goes like this: firstly, it is proven that if $H \subset G$ is a proper ...
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20 views

comparing models of spherical geometry

There more than two models of spherical geometry? 1) one for the half sphere (taking the north pole as center, the boundary is the equator) 2) one for the whole sphere (taking the north pole as ...
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1answer
38 views

$e^{xy}dx \wedge dy$: determine the $1$-form that it induces on $S^1$ and check if the obtained $1$-form respects or not the induced orientation

Consider the $2$-form $e^{xy}dx \wedge dy$ on $\mathbb{R}^2$. Determine the $1$-form that it induces on $S^1$, viewed as the boundary of $B_2$. Check if the obtained $1$-form respects or ...