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Questions tagged [tangent-spaces]

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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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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 $\...
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Tangent Space of the Surface Created from a k-Determined Constraint

For a $k$-determined smooth function $f : \Bbb R^n \to \Bbb R^m ,$ I have a surface $f^{-1}(0)$. Is it true that replacing $f$ with its $k$-jet about zero defines an equivalent tangent space at zero, ...
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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 ...
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2answers
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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
29 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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39 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
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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
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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 ...
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1answer
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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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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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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
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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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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 ...
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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}(...
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1answer
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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, $df:M\rightarrow T^*M$ ...
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1answer
73 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 $\...
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1answer
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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
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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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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
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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
96 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
36 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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1answer
50 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. ...
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4answers
222 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
55 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 ...
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1answer
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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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how can I proove that a tangent space has this direction?

first of all my english isn´t so good so i hope you understand me. I have this homework about tangent spaces and to be honest its pretty complicated when the teacher assume you have an entire course ...
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1answer
26 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 ...
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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. ...
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Tangent space of schemes

Let $X$ be a scheme, $x\in X$ a point. Then Hartshorne defines the tangent space of $X$ at the point $x$ as the dual to the $k(x)$-vector space $\mathfrak{m}_x/\mathfrak{m}_x^2$. The stacks-project ...
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2answers
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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
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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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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=...
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Tangent space of $S^1$

Compute $T_t(S^1)$ in $(\cos(t),\sin(t))$ with parametrization $f(t)=(\cos(t),\sin(t)).$ I have this: I know, $T_tf:T_t\mathbb{R}\to T_{f(t)}S^1$ now $T_tf(\left.\frac{d}{ds}\right|_{s=0}(t+sK))=\...
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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(\...
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1answer
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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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1answer
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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 ...