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Questions tagged [tangent-spaces]

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4
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3answers
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What is the motivation of creating of $T^*_p(\mathbb R^n)?$ How can we visualize covectors?

Question 1 In calculus, we visualize the tangent space $T_p(\mathbb R^n)$ at $p$ in $\mathbb R^n$ as the vector space of all arrows emanating from $p$. What is the motivation of creating of $T^*_p(\...
4
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0answers
127 views

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 ...
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 ...
3
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1answer
53 views

Can a basis of a tangent space be mapped to a basis of another tangent space if the map between the spaces is a homeomorphism and vice versa?

If I have an open subset $U$ of a n-dimensional $C^k-$manifold $M$ and a homeomorphism $f:U \to \Bbb R^n$ (Basically I am talking about a chart $(U,f)$) can I say that under this map a basis of $T_pU=...
2
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4answers
308 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 ...
2
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2answers
29 views

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 ...
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 ...
2
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1answer
57 views

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 ...
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{...
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
votes
1answer
26 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 ...
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 ...
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
60 views

Tangent space of smooth manifold $M=\{(x,x^3,e^{x-1}) : x \in \Bbb{R}\}$ at $(1,1,1)$

What's the tangent space of $M=\{(x,x^3,e^{x-1}): x \in \Bbb{R}\}$ at the point $(1,1,1)$, where $M$ is a manifold of smoothness $C^\infty$. I know how to find the tangent space of a manifold in the ...
1
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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 ...
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 $...
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1answer
31 views

Why $T_{A}M(n) = M(n)$ and why $T_{f(A)}S(n) = S(n)$ and why $M(n)$ & $S(n)$ are manifolds. (Guillemin & Pollack p.23)

Why $T_{A}M(n) = M(n)$ and why $T_{f(A)}S(n) = S(n)$ and why $M(n)$ & $S(n)$ are manifolds? $M(n)$ is the space of all $n x n$ matrices and $S(n)$ is the space of all $n x n$ symmetric matrices. ...
1
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1answer
70 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\...
1
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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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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^...
1
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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 ...
1
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1answer
30 views

Constructing smooth embedding of $M\subseteq \mathbb{R}^n$ into $\mathbb{R}^{n-1}$.

This material is from a class I am taking so some definition might be different from normal sense. So let me define some necessary concepts first and ask question. Definition Let $F:M\...
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1answer
13 views

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. ...
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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1answer
42 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 ...
1
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3answers
84 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 ...
1
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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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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 ...
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1answer
30 views

Definition of derivatives on vector fields on manifolds

While studying the definition of related vector fields for my course in differentiable manifolds, I noticed the following: We gave the following propositions about the derivative of a function $f: M ...
1
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1answer
24 views

Formula for tangent map, proof check

Consider smooth map $f:\mathbb{R}^{n}\to \mathbb{R}$, let $a \in \mathbb{R}^{n}$ be any point, $X \in T_{a}\mathbb{R}^{n}=\mathbb{R}^{n}$ be tangent vector at point $a$. I have probably proven the ...
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1answer
63 views

How does this equality come between a point in $\mathbb R^n$ and combination of operators?

My Doubt:- I understood the proof of $T_p(\mathbb R^n)\simeq \mathcal{D}_p(\mathbb R^n)$. $T_p(\mathbb R^n)$ is a space consists of elements from $\mathbb R^n$. $\mathcal{D}_p(\mathbb R^n)$ is a ...
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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
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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0answers
20 views

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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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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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 $T_x X$ ...
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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
14 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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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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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
62 views

Tangent space basis of $S^3 \times S^3$

I am working with the group morphism $\rho: S^3 \times S^3 \rightarrow SO(4)$ where $\rho(q,r)x = qxr^{-1}$ for $q,r \in S^3$ and $x \in \mathbb{R}^4$ and trying to compute the differential of this ...
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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. ...
0
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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 (...
0
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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 ...
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 ...
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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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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 $\epsilon$: $\text{distance}(x+\epsilon h) = o(\epsilon)$, ...
0
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1answer
18 views

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 ...
0
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0answers
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 $\alpha$; $\frac{\partial s(P,\alpha)}{\partial\alpha}$. ...