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Questions tagged [tangent-spaces]

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2
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1answer
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For any k-dim. subspace $ L$ of $T_p M$, can we find a sub manifold, say $R$, of $M$ containing $p$ s.t $T_p R = L$

Let $M$ be an $n$ dimensional manifold, and $S\subseteq M$ be a k-dim. sub manifold of $M$, where each is in fact a smooth manifold to be precise. We know that $T_p S$ is a k-dim. subspace of $T_p M$....
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1answer
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Determining the linear independence of tangent vectors at a point on the manifold

We define the tangent space at a point, say $x_0$, on the manifold $M$ as the set of all derivations, i.e maps which maps smooth maps from a neighbourhood of $x_0$ to real numbers to real numbers. ...
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0answers
34 views

Are the spaces $T_pM$ and $\mathbb R^n$ homeomorphic?

Let $T_pM$ be the tangent space at a point $p$ in a n-dimensional smooth manifold $M$. In addition, if we assume $(M,g)$ as a smooth Riemannian manifold, then $T_pM$ is a n-dimensional real normed-...
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1answer
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Finding points on a surface $\{ z= f(x, y)\}$ with horizontal tangent plane

Could someone please explain in detail how this is done? For example there is a surface $$M = \{ (x, y, z) : z = x^4 - 4xy^3 + 6y^2 - 2\}$$ and the question is to find the points on $M$ where this ...
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0answers
19 views

Show that if $f$ is constant on a manifold $M$ then $\nabla f$ is orthogonal to the tangent space of each $x \in M$

Let $M \subset R^n$ be a $k$ dimensional manifold. Let $f: R^n \to R$ be a smooth function that satisfies $f(x) = c$, $c \in R$ for every $x \in M $. I need to prove that $\nabla f(x)$ is orthogonal ...
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0answers
15 views

Distance of Manifold and a vector on its tangent space from it.

Let $M \subset R^n$ be a manifold and $x \in M$ a point on it. I want to prove that if $h \in T_xM$ (the tangent space at x) then for every $\epsilon$: $\text{distance}(x+\epsilon h) = o(\epsilon)$, ...
2
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1answer
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Show that if $f$ is a smooth function, $M$ is a manifold and $x$ is a local extremum of $f$ on $M$, then $D_f(x)(v) = 0$ in the tangent space.

Let $M \subset R^n$ be a $k$ dimensional manifold. Let $f: R^n \to R$ be a smooth function. Let $x \in M$ be a local extremum of $f$ on $M$. The task is to prove that $\nabla f_x (v) = 0$ for every $...
2
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1answer
27 views

Quotient by a subring and not an ideal

I'm working towards understanding the Zariski tangent space of a $C^k$ manifold, using this pdf. The author defines $\mathcal{O}^{(k)}_{M,p}$ as the set of germs of $C^k$ functions at $p$, which ...
1
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1answer
71 views

Lie bracket of canonical vectors on tangent space to a point on a manifold is zero.

Let M be a manifold and $T_p(M)$ be the tangent space at $p$, and $\phi$ a local chart around $p$. Let $$\left.\frac{\partial}{\partial\phi^1}\right|_{_p},\ \cdots\ ,\left.\frac{\partial}{\partial\...
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1answer
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Consider the real-valued function $M:=\{(x,y,z)| (2 - (x^2 + y^2)^{1/2})^2 + z^2=1\}$ defined on $\mathbb{R}^3-\{(0, 0, z)\}$.

Show that the manifold $N=\{(x,y,z)\in \mathbb{R}^3|x^2+y^2= 4\}$ is transverse to M. Identify the resulting manifold $N\cap M$. My Attempt: Pardon me for something vacuous, as I am a beginner in ...
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0answers
22 views

KIlling vectors from isometries and orbit spaces

I am currently (trying) to learn more about orbit spaces generated from an isometry group of a manifold. I cannot quite pinpoint what I (don't) understand, so I will try to lay out what I could gather:...
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43 views

Calculating tangent vector of curve s(P,$\alpha$) at given point $\alpha$ = 0. http://yann.lecun.com/exdb/publis/pdf/simard-00.pdf

I am reading one chapter where tangent vector is calculated for the given curve $s(P,\alpha)$ at $\alpha=0$ by differentiating with respect to $\alpha$; $\frac{\partial s(P,\alpha)}{\partial\alpha}$. ...
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0answers
25 views

Understanding the definition of Lie derivative

I'm reading Warner. "Foundations of Differentiable Manifolds and Lie Groups." In p. 69, it gives the definition of the Lie derivative as follows: 2.24 Definition (summerized) Fix a smooth vector ...
3
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1answer
36 views

About the tangent cone and tangent space of an affine variety

Let $X\subset \mathbb{A}^n$ be an affine variety then $X= Z(I)$ is the zero locus of the ideal $I$. In general the tangent cone at $0$ is define as $TC= Z(I^{in})$ where $I^{in}$ is the initial ideal ...
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0answers
33 views

Tangent spaces of two transverse subspaces are transverse subspaces

I am very new to differential geometry and was thrown this very long question: Suppose that two subspaces $V$ and $W$ of $\mathbb{R}^n$ are transverse (so $\text{Span}(V,W)=\mathbb{R}^n$). Let $O$ be ...
1
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1answer
41 views

Why do the properties of a derivation lead to a tangent space of a manifold

From these notes, https://www.dpmms.cam.ac.uk/~md384/neessnmeiwseis.pdf, definition 2.6: A derivation $D$ at $p$ is a mapping $D:X(p) \rightarrow \mathbf{R}$ satisfying $D(\lambda f+\mu g)= \lambda D ...
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Identification of the tangent space of a manifold and the tangent vectors to curves

I'm studying the different definitions of the tangent space for abstract manifolds, and I'm struggling to prove that these abstract concepts reduce to the classical ones when dealing with submanifolds ...
2
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1answer
27 views

How to show that the tangent space as defined by velocity of curves matces intuitive tangent idea?

I'm reading: https://en.wikipedia.org/wiki/Tangent_space Specifically, "Definition as the velocity of curves" and the definition of tangent space at a point as the set of all tangent vectors of ...
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0answers
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Intersection number via tangent spaces

Assume that finite groups $G_1$ and $G_2$ act smoothly on a manifold $M$ in such a way that the fixed point set, $M^{G_1\cap G_2}$, is an oriented closed manifold, $M^{G_1}$ and $M^{G_2}$ are its ...
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2answers
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Germs: Why is it sensible to define a function on a collection of equivalence classes by its action on each element?

I am following Loring W. Tu in his second edition of 'An introduction to manifolds'. Here is a pdf-copy of the book. On page 87 he defines $C^\infty_p(M)$ as the set of germs of $C^\infty$-functions ...
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0answers
31 views

Finding equation of tangent of a vector

hi guys and thanks in advance. I encounter this problem 21 and i couldnt solve it. my answer is so wrong.can anyone guide me on this?
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1answer
18 views

Am I doing this correctly? ( Tangent plane to a surface )

I have this surface, and I'm asked to find the tangent plane at the point $(x,y,z) = (-\alpha, \alpha-1, -\alpha)\quad \alpha>0$ The surface is this one: $(x,y,z)\in\mathbb{R}^3\quad\rvert\quad ...
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0answers
32 views

Dimension of relative tangent space

Let $k$ be an arbitrary field, $X$ be a $k$-Scheme locally of finite type, $x \in X$ a closed point and $\kappa(x)$ its residue field. Question: Is the dimension of of the tangent space $T_x X$ ...
1
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1answer
24 views

Differential map of velocity vector

This is a very basic differential geometry question (please be patient, I am learning) I am given the definition of the differential map of $\phi:M \to N$ as $$d\phi_p(v)(g)=v(g\circ\phi)$$ where $v\...
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0answers
24 views

tangent distance between two handwritten digits

Suppose we have two handwritten images "3"MNIST three and "6"MNIST six. The task is to compute the tangent distance between the two. An explanation at undergraduate level is highly appreciated.
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0answers
15 views

Lie group map whose differential is an isomorphism is a covering map

While trying to read the proof in Fulton and Harris of their “Second Principle,” I ran across something that I do not understand. They seem to claim that if $f: G\rightarrow H$ is a map of Lie groups ...
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0answers
28 views

Infinite dimensional tangent spaces

Let $p \in \mathbb{R}^n$ . It is well-known that there are at least two equivalent definitions of the tangent space at $p$: $T_p \mathbb{R}^n$ is the set of equivalence classes of $C^1$-curves $\...
1
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1answer
43 views

Coordinate basis and coordinate systems

When we introduce coordinate systems, like spherical coordinates, one usually does it with respect to cartesian coordinates. What would be the right way to derive the (for example) spherical ...
2
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2answers
30 views

For which $P,Q \in \text{SO}$ $T_P\text{SO}$ and $T_Q\text{SO}$ are parallel?

I am curious: For which $P,Q \in \text{SO}_n$ does $T_Q\text{SO}_n=T_P\text{SO}_n$ hold? This reduces to the question at the identity,i.e. for which $Q \in \text{SO}_n$, $T_Q\text{SO}_n=T_{Id}\text{...
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1answer
35 views

Need help understanding differential of function

I have encountered the term differential/pushforward many times in the literature, although I cannot seem to understand just what is meant by it. I still cannot seem to understand the definition of ...
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1answer
57 views

Loomis and Sternberg: Tangent Space to a manifold, using equivalence classes; help justifying one step of an argument

I am currently reading through the section in Loomis and Sternberg's Advanced Calculus on Tangent Spaces, but I'm having trouble justifying one step of the argument (shown below). Here's the ...
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3answers
85 views

Understanding Lie algebra of matrix Lie group

In my lecture, we gave a very sloppy (physics people ...) proof of the fact that the Lie algebra $\mathfrak{g}$ of a matrix Lie group $G$ is a subspace of $\text{Mat}_n(\mathbb{F})$. I am not ...
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1answer
37 views

basis of tangent space of a submanifold defined as a graph

Let $f : \mathbb{R}^2 \rightarrow \mathbb{R}$ be a smooth function. Let $G:=\{(x, y, f(x, y)) : x,y \in \mathbb{R}^2\}$ be its graph. Find a basis for $T_pG$ for a $p(x,y,z) \in G$. What I did: I ...
1
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1answer
17 views

smooth curve that is tangent to a 1-form kernel in every point

Let $α = dz - ydx \in Ω^1 (\mathbb{R}^3)$. Prove that $\forall p,q \in \mathbb{R}^3,\ \exists \gamma: [0,1] \rightarrow \mathbb{R}^3$ smooth, such that $γ(0)=p, γ(1) =q$ and $\gamma$ is tangent to $...
2
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2answers
71 views

If $(U,\varphi)$ is a coordinate chart around $p \in M$, where $M$ smooth manifold, then how does $\varphi$ induce coordinates on $T_p M$?

I am studying differential topology and I have some trouble understanding how coordinates are induced on the tangent space at any point. Let $M$ be an $n$-dimensional smooth manifold, and let $p \in ...
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1answer
50 views

Find the tangent space of Ellipsoid $M = \{(x,y,z)|\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1\}$

Find the tangent space of $$M = \{(x,y,z)|\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1\}$$ So I know the formula of tangent space for a manifold represnted by $F$ such that $F=0$: it is $ker (...
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1answer
22 views

Formula for tangent plane to surface given by parametrization

I am aware of how to find an equation of the tangent place to a surface that is given as the graph of a function $z = g(x,y)$. Here one finds a normal vector by essentially taking the partial ...
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0answers
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Question about proof of n-1 form inducing normal unit vector field

Suppose we have a $n-1$ dimensional manifold $M \subset \mathbb{R}^n$ and a non-vanishing $n-1$ form $\omega$ on $M$. This implies the existence of a normal unit vector field on $M$. The proof of ...
0
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1answer
34 views

How/why does the contraction of standard volume form give the canonical form.

$M \subset \mathbb{R}^{N}$ is a (oriented) $n-1$ dimensional submanifold. Suppose $\nu \in T_{p}M^{\bot}$, of length one (a normal unit vector on $M$). How and why does the contraction $\nu_{\neg}(...
2
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1answer
85 views

Tangent space $T_q(df(M))$ as a subspace of $T_q(T^*M)$

I have been asked to describe the tangents space $T_q(df(M))$ as a subspace of $T_q(T^*M)$ where $f\in C^\infty(M)$ and $df$ is a 1-form (or smooth section of $T^*M$). Here, $df:M\rightarrow T^*M$ ...
2
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1answer
84 views

Smooth no-where vanishing form

Does there exist any no-where vanishing smooth $1$-form on $S^2$. I , think there is such one. For example, consider the smooth $1$-form $\omega=dx+dy+xdz$ on $\Bbb R^3$ consider the pull-back of $\...
2
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1answer
58 views

Tangent Spaces (Algebraic Geometry)

I'm in my algebraic geometry class and I have the definition of the space tangent to some variety $W=V(F_1,F_2,...)$ as the degree-1 components of $F_1,F_2,...$ . We then introduce the differential at ...
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1answer
33 views

Matrix associated of an application between tangent spaces

Let $M$ be a differential manifold and $X$ a vector field over $M$ s.t. $X(p) = 0$ for some $p \in M$. Let be $\phi_p : T_p(M) \to T_p(M)$ given as $$\phi_p(v) := [Y,X](p),$$ being $Y$ another vector ...
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0answers
39 views

Tangent space for the differentiable manifold $S^1$

Since $S^1$ is a compact 1-dimensional regular submanifold in $\mathbb{R}^2$ (it's $S^1 = f^{-1}(1)$ for $f : \mathbb{R}^2 \to \mathbb{R}$ given as $f(x,y) = x^2+y^2$), we can find the tangent space ...
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2answers
70 views

Push-forward of inverse map

If I define the inverse map in a Lie group $G$ as, $$i: G \rightarrow G,\quad i(g) = g^{-1}, \forall g \in G \tag1$$ I think that the associated push-forward would be, $$i_*: T_gG \rightarrow T_{g^...
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1answer
107 views

Approximating the tangent vector in a phase space (or state space) reconstruction

I am investigating an application of differential geometry in experimental dynamical systems. Given a 1D time series (e.g., one that has been experimentally obtained), $x(t)$, I am considering the ...
0
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1answer
43 views

Tangent Space and Charts

I don't come from a Differential Geometry background but I have been trying to read a bit about Lie algebras. I am using Humphrey's as my main source but just to get a glimpse of the correspondance ...
0
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1answer
55 views

Calculating the derivative of a mapping $\varphi: S^2 \rightarrow S^2$

I have recently learned about tangent spaces and derivatives in the context of manifolds and I am having a hard time solving the following exercise: Let A be a $3\times3$ orthogonal matrix. ...
2
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4answers
309 views

How do I find a tangent plane without a specified point?

I was having a problem finding the points on $z=3x^2 - 4y^2$ where vector $n=<3,2,2>$ is normal to the tangent plane. How do we calculate the tangent plane equation without a specific point to ...
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1answer
72 views

How to recover the tangent space from the metric

This seems such an elementary question, but I cannot see how to do this. Say that you are being given a metric (locally of course): $$ g =ds^2 = g_{\mu \nu} dx^\mu dx^\nu $$ Since the metric encodes ...