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Questions tagged [tangent-spaces]

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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
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
58 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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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 ...
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1answer
35 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}(...
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1answer
24 views

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 ...
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 ...
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0answers
20 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
23 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
33 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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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
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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0answers
45 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 $\epsilon$: $\text{distance}(x+\epsilon h) = o(\epsilon)$, ...
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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 $\alpha$; $\frac{\partial s(P,\alpha)}{\partial\alpha}$. ...
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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 ...
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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 ...
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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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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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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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23 views

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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0answers
50 views

Confusion about tangent spaces

Hartshorne defines the Zariski tangent space at a point $x$ of a scheme $X$ to be $(\mathfrak{m}_x/\mathfrak{m}_x^2)^*$, where the star $*$ means taking the dual, i.e. homomorphisms $\mathfrak{m}_x / \...
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0answers
34 views

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))=\...