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Questions tagged [tangent-spaces]

This tag is for questions regarding to the tangent space, the linear space that best approximates an object at a given point. Intuitively, the tangent space $ T_p(M)$ at a point $ p$ on an $ n$-dimensional manifold $ M$ is an $ n$-dimensional hyperplane in $ {\mathbb{R}}^m$ that best approximates $ M$ around $ p$, when the hyperplane origin is translated to $ p$.

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Tangentbundle of submanifold of $\mathbb{R}^n$

Let $M\subseteq \mathbb{R}^n$ be a $k$-dim submanifold of $\mathbb{R}^n$, and I want to prove that the tangent bundle of $M$ is a submanifold of $\mathbb{R}^n$. The idea is: Since $M$ is a submanifold ...
Gao Minghao's user avatar
1 vote
1 answer
64 views

Tensor Notation with Basis in Differential Geometry

Let's say we have two smooth riemannian manifolds $\mathfrak{B}$ and $\mathfrak{S}$ and with coordinates $X^A$ on $\mathfrak{B}$ and $x^a$ on $\mathfrak{S}$, with $A,a \in \{1,2,3\}$ Let's now assume ...
Noiv's user avatar
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Are planes through the origin the only position vector self-tangent surfaces?

Are there any surfaces in $\mathbb{R}^3$ (other than planes through the origin) such that each position vector lies in the respective tangent plane at that point? If the surface is given by say $\phi:...
Derso's user avatar
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1 vote
2 answers
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The action of $SL(n,\Bbb{R})$ on the tangent space of $SL(n,\Bbb{R})/SO(n)$.

SETUP. It is a standard result that $\text{GL}(n,\Bbb{R})/O(n)$ is isomorphic to the set $P'$ of positive definite $n\times n$ matrices, as manifolds: the basic idea is that $\text{GL}(n,\Bbb{R})$ ...
SomeCallMeTim's user avatar
3 votes
0 answers
77 views

Tangent space to locus of varieties containing a given subvariety (Hilbert schemes)

Suppose that I have a variety $X$ in $\mathbb{P}^r$ of a given type. Then I know that the tangent space to the irreducible component $\mathcal{H}_X$ of the Hilbert scheme containing $[X]$ can be ...
maxo's user avatar
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1 answer
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Curl of a vector field belongs to the tangent space of a manifold

While studying Analysis on Manifolds in Elon Lages book, in the Chapter of Stokes Theorem there was this problem: Let U $\subseteq$ $\mathbb{R}^3$ be a open set and consider $F: U \rightarrow \mathbb{...
werner_math's user avatar
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45 views

Is the derivative at a point x of a smooth real-valued map linear?

I am currently reading "Differential Topology" by Victor Guillemin and Alan Pollack. They are in the process of explaining the preimage theorem in terms of a set of common zeroes (to show ...
Tosca's user avatar
  • 41
1 vote
1 answer
31 views

Page 67 Lee smooth manifold - transition map of charts on tangent bundle clearly smooth?

Lee wrote the following on page 67 on his Introduction to Smooth Manifolds The Tangent Bundle Now suppose we are given two smooth charts $(U, \varphi)$ and $(V, \psi)$ for $M$, and let $\left(\pi^{-1}...
wsz_fantasy's user avatar
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0 answers
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Outward unit normal of the Superegg

I´m working with a tube-like object in 3D, which consists of a cylinder of radius $B$ at center $(x_0, y_0)$ with height $L-H$, this is glued together from above with a superegg with the same radius ...
oli H.'s user avatar
  • 329
2 votes
2 answers
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Second partial derivatives as tangent vectors on a smooth manifold

Let $M$ be a Riemannian manifold along with a chart $(U,x)$. I'm wondering if second partial derivatives can also be seen as tangent vectors. I'm assuming this is not the case, since for example one ...
John Doe's user avatar
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Tangent space of a $(d-1)$-sphere

I am trying to verify the following statement: Let $\mathbb{S}^{d-1} \subseteq \mathbb{R}^d$. For any $v\in T_o ~ \mathbb{R}^d$ with $o$ being the origin $(0,0,...,0)$ of $\mathbb{S}^{d-1}$, $\exists ...
SCh's user avatar
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0 answers
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Canonical Basis of Tangent Space of a Manifold regarding Charts

Consider the upper half of the sphere, $\mathbb{S}^2_+:=\{p\in S^2\mid p_3>0\}\in\mathbb{R}^3$and the charts $(\mathbb{S}^2_+, x=\mathcal{i}_N), (\mathbb{S}^2_+, y=\pi_N)$ where $\mathcal{i}_N$ is ...
Lu1998's user avatar
  • 27
1 vote
1 answer
44 views

Geodesic tangent space is a vector space?

I have very little knowledge of Differential Geometry and I'm stuck while reading about General Relativity. Consider defining something called a null geodesic tangent space, in analogy with the ...
SCh's user avatar
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0 answers
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proof that the wasserstein space is no manifold

This is my first question on this platform, I appreciate any suggestions on how to improve my question. why is the Wasserstein space no manifold and in which way is its structure somehow similar to a ...
arm's user avatar
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0 answers
132 views

$H^{1/2}(\partial\Omega)$-regularity of tangent vector field defined on smooth boundary

Suppose that $\Omega$ is a bounded, smooth, simply connected domain in $\mathbb{R}^3$. Let $p(x):\partial \Omega \to \mathbb{S}^2$ be a vector field such that it lies on the tangent plane of $x \in \...
mnmn1993's user avatar
  • 395
2 votes
1 answer
50 views

Tangentspace of product of manifolds

I am currently trying to prove that for differentiable manifolds $M \subset \mathbb{R}^m$ and $N \subset \mathbb{R}^n$. It holds that: $T_{(x,y)} (M \times N) = T_x M \times T_y N$ for arbitrary $x \...
user007's user avatar
  • 615
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0 answers
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Computing the tangent space of the orbit of a gauge group action at a connection

Let $E\to M$ be a smooth real vector bundle, and let $\mathfrak{G}$ be the group of smooth bundle automorphisms. (The Lie algebra of $\mathfrak{G}$ is the space $\Omega^0(\text{End}(E))$.) For a ...
blancket's user avatar
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1 vote
1 answer
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Confusion about one pushforward calculation on $\mathbb{S}^{n-1}$

Let $\mathbb{S}^{n-1} \subset \mathbb{R}^n$ be the standard sphere and let $$ F \colon \{ (p, r, X) \colon \ p \in \mathbb{S}^{n-1}, \ r > 0, \ X \in T_p \mathbb{S}^{n-1}, \ |X| = 1\} \to \mathbb{S}...
tsnao's user avatar
  • 320
1 vote
1 answer
50 views

About Differentiability in geometric definition of tangent space

I have a question about the definition of the tangent space which confuses me a lot. For the geometric definition of the tangent space $T_xM$ at the point x for a differentiable manifold M we have ...
GG314's user avatar
  • 114
1 vote
0 answers
42 views

inequality in tangent space

It is well-known that in $\mathbb{R}^N$ the following vector inequality holds : $$(|x|^{N-2}x-|y|^{N-2}y).(x-y)\geq2^{2-N}|x-y|^N$$ for $N\geq2$ and $x,y\in\mathbb{R}^N$. My question is whether the ...
am_11235...'s user avatar
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1 vote
1 answer
98 views

Canonical isomorphism between tangent space and translation vector space of affine space

I've a question about the following. Take an affine space $(E,V)$ and consider the tangent space $T_aE$ at a point $a$. From John Lee book "Introduction to smooth manifolds" chapter 3 there ...
Carlo C's user avatar
  • 25
1 vote
0 answers
53 views

Metric choice in the tangent space of a Riemannian manifold obtained through the Log map

After using the Log map, as defined in this paper Riemannian approaches in Brain-Computer Interfaces: a review (Section III. A page 2&3), to project points from the manifold onto the tangent space ...
user19402204's user avatar
0 votes
1 answer
38 views

Properties of Lie subalgebra complements [closed]

I have a Lie algebra that breaks up into as subalgebra $B$ and its complement $\overline B$ (dividing the set of Lie generators into mutually exclusive subsets $B$ and $\overline B$ such that $B$ is a ...
Bob McElrath's user avatar
1 vote
1 answer
38 views

Can $C_p^\infty(\mathbb{R}^n)$ be considered as a dual space of $T_p\mathbb{R}^n$, where $p\in \mathbb{R}^n$?

warning: This may be a stupid question, because I am poor at differential euqations $v_p\in T_p\mathbb{R}^n$ is a linear function over $C_p^\infty (\mathbb{R}^n)$, where $C_p^\infty (\mathbb{R}^n)$ ...
Richard Mahler's user avatar
3 votes
1 answer
62 views

How to determine a basis for the tangent space given a local trivialization.

I heard the following: For a smooth submanifold $M \subseteq \mathbb{R}^n$, given a local trivialization one can easily find a basis for the tangent space. I want to know how. So first I should ...
Peter's user avatar
  • 476
1 vote
1 answer
116 views

Differentials vs one-forms

When one is first introduced to a differential of a real-valued function, it is defined as a linear function that takes tangent vectors to real numbers. That is, differential is a linear functional on ...
Максим Неважно's user avatar
4 votes
1 answer
50 views

On a Universal Property for the Tangent Space.

A time ago, I was asking if there exists an universal property of the tangent space and what it says about any construction of it. I've found the definition maded in Tammo Dieck's book of Algebraic ...
Paulo Estêvão's user avatar
1 vote
0 answers
39 views

How does the differential of a complex endomorphism of $\mathbb C^n$ act on $\frac{d}{dz^i}$ and $\frac{d}{d\bar{z}^i}$?

I caught myself up in a seemingly simple question. Suppose $f: \mathbb C^n \rightarrow \mathbb C^n$ is a $\mathbb C$-linear map given by a matrix $A \in M_n(\mathbb C)$. I am trying to figure out how ...
rosecabbage's user avatar
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2 votes
1 answer
94 views

What is the maximal l for a $C^{l}$-vector field on a $C^{k}$-manifold?

Given a $C^{k}$-manifold what is the maximal $l$ that allows to have a good definition of a $C^{l}$-vector field on that manifold? In some textbooks is given a definition with $l=k-1$ and restriction $...
Danilo Lombardo's user avatar
1 vote
0 answers
35 views

A doubt regarding the orthogonality of the partials defined on a tangent space of an abstract manifold.

Suppose $M$ is a manifold of dimension $n$.Let $p\in M$ and let $\psi:U\subset \mathbb R^n\to M$ be a local parametrization on $M$ near $p$.So,$\psi:U\to \psi(U)$ where $\psi(U)$ is open in $M$ and $p\...
Kishalay Sarkar's user avatar
1 vote
0 answers
70 views

How do we define the map on Zariski tangent space in the infinite case?

Reading Liu, I got quite confused about what the "canonical map" $T_{f,x}:T_{X,x}\to T_{Y,f(x)}\otimes_{k(f(x))}k(x)$ should be for a map of general schemes $f:X\to Y$. Reading several MSE ...
FShrike's user avatar
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2 votes
2 answers
149 views

Tangent space of $\mathbb{P}(V)$

I encountered this problem studying the local period map and I'm wondering how to solve it. I would like to prove that, given $V$ a complex vector space and $W \subseteq V$ a one-dimensional subspace, ...
WindUpBird's user avatar
0 votes
1 answer
52 views

Why do derivations take only smooth inputs?

In ‘An Introduction to Manifolds’ Tu introduces a concept of derivations at a point to define tangent spaces. The one thing I do not quite get is why derivations act only on smooth function. Why can’t ...
Максим Неважно's user avatar
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0 answers
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Pushforward of a derivation

I am trying to compute the pushforward of a tangent vector as a derivation. Here are my definitions: Let $v\in \left.(TM)\right|_U$ be a tangent vector at the point $p$. Given some coordinates $(\phi,...
Bedge's user avatar
  • 241
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0 answers
39 views

A question on alternating product and $SL_n(R)$

I am reading Fulton and Harris representation theory. In the section 8.2 - Examples of Lie Algebras, while calculating the Lie algebra of $SL_n(\mathbb R),$ the author tell that by definition $$ A_t(...
Eloon_Mask_P's user avatar
1 vote
0 answers
27 views

Compute the tangent map on a Lie group

Let $G$ be a real Lie group with identity $e$. Let $x_0\in G$ and $U\subset G$ be a submanifold containing $x_0$. Consider the map $$\varphi:G\times U\rightarrow G,\quad \varphi(g,u):=gug^{-1}.$$ We ...
youknowwho's user avatar
  • 1,499
3 votes
0 answers
86 views

Properties of tangent functor

When I was learning differential geometry, I was told that differential of a function can be viewed as a functor, i.e. sending manifold into tangent space and function into its differential. ...
Liam's user avatar
  • 333
2 votes
2 answers
118 views

What is a transversal intersection? Can it be explained without tangent spaces and tangent bundles?

What is a transversal intersection? Can it be explained without tangent spaces and tangent bundles? Background: Transversal intersection was used to explain If the interior of two convex manifolds ...
SRobertJames's user avatar
  • 4,450
0 votes
1 answer
74 views

Tangent plane orthogonal to a vector

I have some trouble to do this excercise: Given $S \subset \mathbb{R}^3$ the implicit surface defined as $$ S=\left\{(x, y, z) \in \mathbb{R}^3 \left\lvert\, x^2+x z+y z+\frac{1}{2} z^2=1\right.\right\...
Matias's user avatar
  • 85
1 vote
1 answer
36 views

Is the range of a vector field $\bigcup_{p\in\mathbb{R}^n}\mathbb{R}_p^n$? ("Calculus on Manifolds" by Michael Spivak.)

I am reading "Calculus on Manifolds" by Michael Spivak. The author defined a vector field as follows: To be precise, a vector field is a function $F$ such that $F(p)\in\mathbb{R}_p^n$ for ...
佐武五郎's user avatar
  • 1,138
0 votes
0 answers
34 views

I think $\nabla=\sum_{i=1}^n D_i\cdot (e_i)_p$ is correct. ("Calculus on Manifolds" by Michael Spivak)

I am reading "Calculus on Manifolds" by Michael Spivak. The author defined $\langle F,G\rangle$ as $\langle F,G\rangle(p)=\langle F(p), G(p)\rangle$ for vector fields $F$ and $G$. But I ...
佐武五郎's user avatar
  • 1,138
2 votes
2 answers
109 views

Why do we need a tangent space? Why do we need a vector field? ("Calculus on Manifolds" by MIchael Spivak)

I am reading "Calculus on Manifolds" by Michael Spivak. Why do we need a tangent space? Why do we need a vector field? To be precise, a vector field is a function $F$ such that $F(p)\in\...
佐武五郎's user avatar
  • 1,138
0 votes
1 answer
43 views

Definition of Tangent Map is compatible with the definition of a Tangent Vector

A tangent vector at a point p $\in$ M can be defined as an equivalence class of curves $\gamma:I\rightarrow M$ by the equivalence relation \begin{align} \gamma_1 \sim \gamma_2 \Leftrightarrow \frac{d}{...
Aralian's user avatar
  • 119
0 votes
1 answer
59 views

Computing $ι^∗(dy^j)_p( \frac{\partial}{\partial x^l}_p)$ and $ι^∗(dz^k)_p( \frac{\partial}{\partial x^l}_p)$on $Graph(F) = \{(a, F(a)); a \in V \}$

Let $V \subseteq \Bbb R^m$ open, and $F : V \to \Bbb R^n$ smooth. Consider the graph of $F$ $\operatorname{graph}(F) = \{(a, F(a)); a \in V \}$ which is an embedded submanifold of $\Bbb R^m \times \...
some_math_guy's user avatar
1 vote
1 answer
90 views

Hairy Ball Theorem on $TS^1$ and $TS^2$

Edit: It seems my language was not correct. By "orientable", I mean there are no non-zero sections of the tangent bundle, i.e. the sphere fails the Hairy Ball Theorem, while the circle does ...
Fly by Night's user avatar
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0 votes
0 answers
38 views

Vertical vector space = tangent space to a fiber

Lee's Introduction to Riemannian manifolds (2nd ed.) introduces the vertical tangent vector space as follows (Page 21): Here, $\tilde{M}_p = \pi^{-1}(p)$ for $p\in M$. Suppose $\tilde{M}$ and $M$ are ...
Kaira's user avatar
  • 1,565
1 vote
0 answers
94 views

Tangent space and derivations on a Banach manifold

When dealing with a finite $d$-dimensional manifold $M$, one can define the tangent space $TM|_p$ of a manifold on a point $p \in M$ in different (but equivalent) ways, based on (at least) the ...
Pedro G. Mattos's user avatar
3 votes
1 answer
105 views

Tangent map vs. differential on manifolds

Let $E, E'$ be normed vector spaces, $A \subseteq E$ an open set and $f: A \to E'$ a differentiable map. For each point $x \in A$, the differential of $f$ is a linear transformation $Df|_x: E \to E'$. ...
Pedro G. Mattos's user avatar
-4 votes
1 answer
219 views

Why am I wrong about the tangent bundle? [duplicate]

What is gained by insisting on the distinction between tangent spaces at different points and double-tagging them in the construction of the tangent bundle? What specifically am I missing by ...
R. Burton's user avatar
  • 5,040
0 votes
1 answer
92 views

Is there a way to convert an element of $\mathfrak{so}(3)/\mathfrak{so}(2)$ into a "geometric" tangent vector on $S^2$?

From my understanding, the tangent space at the identity of the homogeneous space $\rm SO(3) / \rm SO(2)$ is just the quotient space $\mathfrak{so}(3) / \mathfrak{so}(2)$. An element in this quotient ...
Max0815's user avatar
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