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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2
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0answers
19 views

Show that $T_{(p,q)}(M\times N)\cong T_p M\oplus T_q N$. Tangent space to the product manifold.

I want to check my arguments for the following proof. Let $M$ and $N$ be smooth manifolds where $\dim M=n$, $\dim N = m$, and $\pi_1:M\times N\to M$, $\pi_2:M\times N\to N$ be corresponding projective ...
0
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1answer
39 views

Is it possible to define a vector field of non-coordinate basis vectors?

I'm studying non-coordinate basis of (pseudo-)riemannian manifolds and I found the following definition from Nakahara - Geometry, topology and physics: a non-coordinate basis $\{\hat{e}_\alpha\}$ is ...
3
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1answer
41 views

How to show that $T_{(1,0)}\mathbb S^1 \cong \operatorname{span}(\{e_2\})$?

I want to show that $T_{(1,0)}\mathbb S^1 \cong \operatorname{span}(\{e_2\})$ using the stereographic chart and using the definition that $T_xM$ is the set of velocity vectors $v$ where each vector $v$...
0
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0answers
19 views

Material derivative on moving boundary

Let $T_t$ be a $C^1$-diffeomorphism on $\mathbb R^d$, $$v_0(x):=\left.\frac{\rm d}{{\rm d}t}T_t(x)\right|_{t=0}\;\;\;\text{for }x\in\mathbb R^d,$$ $\Omega$ be a $k$-dimensional embedded $C^1$-...
0
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1answer
42 views

If $M_i$ is a submanifold with $M_1\subseteq M_2$, what's the pushforward of the restriction $\left.f\right|_{M_1}$ of a map $f:M_2\to\mathbb R$?

Let $d\in\mathbb N$, $k_i\in\{1,\ldots,d\}$, $M_i$ be a $k_i$-dimensional embedded $C^1$-submanifold of $\mathbb R^d$ with boundary and $f_i:M_i\to\mathbb R$. Assume $M_1\subseteq M_2$ and $f_1=\left....
2
votes
2answers
78 views

Is always possible define a coordinate basis for a smooth manifold?

The coordinate basis or holonomic basis for a differentiable manifold $\mathcal{M}$ is a set of basis vector fields $\{e_\mu\}$ definited in each point $P\in \mathcal{M}$ with the local condition $$ [...
2
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0answers
21 views

Lipschitz function is differentiable at a point iff its tangent set is a k-dimensional plane

I'm reading a proof of Hadamard-Perron theorem from Katok's Introduction to the Modern Theory of Dynamical Systems. I'm having problems with the following part. Let $\varphi:\mathbb{R}^k\to\mathbb{R}^{...
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0answers
20 views

If a function has a local $C^1$-extension, is it locally continuous?

Let $E_i$ be a $\mathbb R$-Banach space, $\Omega_i\subseteq E_i$, $x_1\in\Omega_1$ and $f:\Omega_1\to\Omega_2$ be $C^1$-differentiable at $x_1$ (see below). Question 1: Can we show that $f$ is ...
4
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1answer
66 views

Differential in terms of $T_pX \simeq \left(\mathfrak{m}_p/ \mathfrak{m}_p^2\right)^\ast$

I am studying different definitions of the tangent space to a manifold $X$ in a point. When we identify $T_pX \simeq \left(\mathfrak{m}_p/ \mathfrak{m}_p^2\right)^\ast$, how can we express the ...
3
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0answers
36 views

Tangent space of an upper half-space and the relation of $T_x\:M$ and $T_x\:\partial M$, for $x\in\partial M$, in general

Let $d\in\mathbb N$, $k\in\{1,\ldots,d\}$ and $M$ be a $k$-dimensional embedded $C^1$-submanifold of $\mathbb R^d$ with boundary. What is the relation between$^1$ $T_x\:M$ and $T_x\:\partial M$ for $...
1
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0answers
32 views

A tangent vector of a manifold $M$ with boundary is either inward/outward pointing or tangential to $\partial M$

Let $d\in\mathbb N$, $k\in\{1,\ldots,d\}$ and $M$ be a $k$-dimensional embedded $C^1$-submanifold of $\mathbb R^d$ with boundary$^1$. Say that $(I,\gamma)$ is a $C^1$-curve on $M$ through $x\in M$ if $...
0
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0answers
16 views

If $ϕ$ is a boundary chart of a $k$-dimensional submanifold $M$ with boundary, then $(T_xϕ)v\in\mathbb R^{k-1}×\{0\}$ for all $v\in T_x\:\partial M$

Let $k\in\{1,\ldots,d\}$ and $M$ be a $k$-dimensional embedded $C^1$-submanifold of $\mathbb R^d$ with boundary, $(\Omega,\phi)$ be a $k$-dimensional $C^1$-chart$^1$ of $M$ and $$U:=\phi(\Omega\cap\...
0
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2answers
138 views

If $(T_t)_{t\ge0}$ is a flow on a submanifold $\Omega$ with boundary with velocity $v$ satisfying $\langle v,\nu_{∂Ω}\rangle=0$, then $T_t(∂Ω)⊆∂Ω$

Let $\tau>0$; $d\in\mathbb N$; $v:[0,\tau]\times\mathbb R^d\to\mathbb R^d$ be Lipschitz continuous in the second argument uniformly with respect to the first with $v(\;\cdot\;,x)\in C^0([0,\tau],\...
0
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0answers
21 views

Tangent space of the union of affine varieties

I want to prove that given $X, X_1, X_2$ affine varieties, if $X = X_1 \cup X_2$ and $x \in X_1 \cap X_2$, then the tangent spaces $T_{x, X_1}, T_{x, X_2}, T_{x, X}$ satisfy $$T_{x, X_1} + T_{x, X_2} \...
2
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0answers
67 views

Motivation for defining tangent vectors with derivations and why they should act on $f\in C^\infty(M)$

I'm revisiting the definition for tangent spaces in Lee's Introduction to Smooth Manifolds and I'm trying to convince myself why we might define tangent vectors as derivations at a point $p\in M$: ...
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0answers
23 views

Does image of lipschitz function have continuous tangent? [closed]

Let $f: \Bbb R^m \rightarrow \Bbb R^n$ be some lipschitz function. And let $A$ be $f(\Bbb R^m)$. We define a function $x \rightarrow \theta_x$ that sends $x$ to the tangent field of $x$ at $A$. Is the ...
3
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1answer
150 views

Definition of the tangential gradient

Let $d\in\mathbb N$ and $M\subseteq\mathbb R^d$ be bounded and open such that $\partial M$ is of class $C^1$ (i.e. a $(d-1)$-dimensional embedded $C^1$-submanifold of $\mathbb R^d$). If $f:\partial M\...
0
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1answer
49 views

Definition of the tangent space for a submanifold with boundary in terms of regular curves

Let $d\in\mathbb N$, $M\subseteq\mathbb R^d$ and $x\in M$. $\gamma$ is called curve on $M$ through $x$ if $\gamma:I\to M$ for some nontrivial interval $I\subseteq\mathbb R$ with $0\in I$ and $\gamma(0)...
2
votes
1answer
79 views

differentials and tangent space of a fibre

The setup I have is as follows: Let $f: X \to Y$ be a morphism of non-singular $n$-dimensional varieties (separated reduced irreducible scheme of finite type over $k$) over $k$ an algebraically closed ...
1
vote
1answer
86 views

How can we show that this normal field is “outward pointing”?

Let $d\in\mathbb N$, $\alpha\in\mathbb N$ and $M$ be a $d$-dimensional embedded $C^\alpha$-submanifold of $\mathbb R^d$ with boundary. How can we show that there is a unique $\nu_M:\partial M\to\...
0
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0answers
30 views

Extending the normal field of a $d$-dimensional submanifold $M$ of $\mathbb R^d$ with boundary to an open neighborhood of $\partial M$

Let $d\in\mathbb N$ $\alpha\in\mathbb N$ $M$ be a $d$-dimensional embedded $C^\alpha$-submanifold of $\mathbb R^d$ with boundary $\operatorname{Bd}(M)$ and $\partial M$ denote the topological and ...
0
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1answer
22 views

Why $I^2$ a vector space where $I$ is the space of differentiable functions vanishing at a point $x$

I am reading the following wikipedia article on tangent spaces, in particular, this subsection on the definition via the cotangent space. Here is a paraphrasing of the first two sentences of the first ...
2
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1answer
51 views

Identification of injective linear maps with image and inclusion

Suppose $S$ is an immersed submanifold of $M$. Let $\iota: S\hookrightarrow M$ be the inclusion map. Since it is an immersion, at each $p\in S$, $\iota_{*}:T_pS\rightarrow T_pM$ is injective. Hence we ...
0
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0answers
14 views

Show that the outer unit normal field is contained in the null space of the adjoint of the tangential Jacobian

Let $\Omega\subseteq\mathbb R^d$ be bounded and open and $\partial\Omega$ be of class $C^1$. Denote the tangential Jacobian and garient by ${\rm D}_{\partial\Omega}$ and $\nabla_{\partial\Omega}$, ...
0
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1answer
40 views

'Classical' Infinitesimals and Tangent Spaces

I do not know much differential geometry, and was led to this question from complex dynamics. It seems that it is often possible to reason 'infinitesimally' about maps between tangent spaces. For ...
1
vote
1answer
99 views

Confusion about how tangent vectors relate to vector fields in Differential geometry

TL;DR: I'm confused about some very basic definitions in Differential geometry, namely how tangent vectors and vector fields are related to each other. Let $M$ be smooth manifold. My professor ...
0
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2answers
87 views

How can a vector field act on a Lie Algebra element?

We have the definition of a vector field as a smooth section of the tangent bundle $$X:P\longrightarrow TP,$$ where $(TP,\pi',P)$ is the tangent bundle over the total space of the principal G-bundle $(...
1
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0answers
37 views

$S = \{ A \in M_{n}(\mathbb{R}) : 0 <\operatorname{rank}(A) = p < n \}$ is a graph of a $C^{1}$ function

Consider $S = \{ A \in M_{n}(\mathbb{R}) : 0 < \operatorname{rank}(A) = p < n\}$, where p is a fixed integer. Suppose that $n = 2$ and $p = 1$. Show that, locally, $S$ is the graph of a real $C^...
1
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1answer
39 views

Tangent space of projection operators in $\mathbb{R}^3$

In a problem I'm currently tackling (not related to the question) the map $f : S^2 \times \mathbb{R}^3 \to \mathbb{R}^3$ is defined as $$ (d,v) \to \langle v,d\rangle d = dd^T v $$ ($S^2$ is the unit ...
0
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1answer
22 views

The Explanation of these steps

I was following a lecture on Tangent Spaces, where I find expressions as: $$(f\circ\gamma\circ\mu)'(0) = (f\circ\gamma)'(\mu(0)).\mu'$$ And in some other place, I find: $$((f\circ x^{-1})\circ(x\circ\...
1
vote
1answer
62 views

Why is it important the manifold has codimension $1$ in order to prove this identity for $\operatorname{div}fV$ on $\partial M$?

I've seen the following claim in some lectures notes which let me think that I might have a major misunderstanding: The claim is that if $M$ is an embedded submanifold of $\mathbb R^d$ with boundary ...
2
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0answers
37 views

Show an identity for the Laplace-Beltrami operator

Let $\partial M$ denote the boundary of a $k$-dimensional embedded $C^1$-submanifold $M$ of $\mathbb R^d$, $T_x(\partial M)$ and $N_x(\partial M)$ denote the tangent and normal field of $\partial M$ ...
2
votes
1answer
130 views

Characterization of the tangent space of the boundary of an embedded submanifold of $\mathbb R^d$ with boundary

Let $M$ be a $k$-dimensional embedded $C^1$-submanifold of $\mathbb R^d$ with boundary, i.e. $M$ is locally $\mathcal C^1$-diffeomorphic$^1$ to $\mathbb H^k:=\mathbb R^{k-1}\times[0,\infty)$, $$T_xM:=\...
0
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0answers
29 views

Prove intersection of 2-manifolds of $R^3$ is 1-manifold

I'm currently struggling with the following problem: Let $M_1, M_2 \subset \mathbb{R}^3$ be two-dimensional submanifolds of $\mathbb{R}^3$ with $M_1 \cap M_2 \neq 0$, s.t. for all $x \in M_1 \cap M_2$...
0
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1answer
36 views

Orthonormal bases and orthonormal frames

Let $\{E_1,...E_n\}$ be a set of vector fields defined on a domain $U \subset \mathbb{R}^n$ such that for all $p \in U$, $\{E_1(p),...,E_n(p)\}$ is an orthonormal basis for $T_pU$ (tangent space at $p ...
2
votes
0answers
45 views

Wave maps, second energy inequality

Sorry in advance for the long setup: Let $u: V \rightarrow M$, where $(V,g)$ is a Lorentzian manifold and $(M,h)$ is a Riemannian manifold. Also $V=S \times R$, where $S$ is an $n$-dimensional ...
1
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0answers
52 views

Commutation of covariant derivatives

Let $u: V \rightarrow M$, where $(V,g)$ is a Lorentzian manifold and $(M,h)$ is a Riemannian manifold. Also $V=S \times R$, where $S$ is an $n$-dimensional orientable smooth manifold. Now i $\...
0
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0answers
15 views

Tangent space of a Principal bundle

Suppose we have a Principal bundle with a Lie group G in fiber. It is known that through the trivializations it can locally be expressed as a product of the base space M and the group in the fiber. ...
2
votes
1answer
41 views

How are tangent space basis vectors affected under a change of coordinates?

I'm studying 'Core Principles of Special and General Relativity' by Luscombe - the chapter on tensors. Quoting: Consider an arbitrary three-dimensional coordinate system where point $P$ is at the ...
0
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1answer
24 views

Fundamental form of almost complex manifold is $(1,1)$-form

Let $M$ be an almost complex manifold with almost complex structure $J$, a compatible Riemannian metric $g$ and fundamental form $\omega$. Consider the eigenspaces $T_p^{1,0}M=\{v\in T_pM\otimes\...
1
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1answer
53 views

Does a Lie derivative of a vector field involve subtracting vectors from different spaces?

Addition (and subtraction) is not by default defined for vectors in different spaces, even if those vector spaces are isomorphic (it is possible to define addition, but there are many ways to define ...
0
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1answer
53 views

How are Gerstner wave normals derived?

I'm looking at this GPU gems article on water rendering. It gives the following tangent space vectors. I understand the normals N are calculated as the cross product B x T, but I can only seem to ...
0
votes
1answer
22 views

Quaternion interpolation with SQUAD: Tangent issue

For interpolating between keyframes in an animation path I make use of SQUAD(q1,q1_t,q2_t,q2) function. q1 and q2 beeing the points to interpolate and q1_t and q2_t ...
1
vote
1answer
39 views

Partial Derivatives on Manifolds as Derivations

first time poster, finally decided to take the plunge and not just lurk anonymously. I'm just an experimental physicist with a desire to know some math beyond my few semesters of abstract algebra. ...
1
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0answers
53 views

Proving the union of tangent spaces of $M\subset\Bbb{R}^{p}$ has measure zero if $2\dim M < p$

Let $M\subset \Bbb{R}^{p}$ a surface of class $C^2$. I need to prove that, if $2\dim M < p$, so the set $$X=\bigcup_{x\in M} T_x M$$ has measure zero in $\Bbb{R}^{p}$. My definition of "measure ...
1
vote
1answer
37 views

Tangent space of a group of diffeomorphisms

In a paper I was reading the following result was used: Let $\Gamma= Diff^{+}([0,1]^2)$ be the set of all boundary preserving diffeomorphisms on $[0,1]^2$, then the Tangent space $\mathcal{T}_{\gamma_{...
1
vote
2answers
31 views

How to obtain real vector from abstract tangent vector in the case of the manifold $\mathbb R^n$

I know that for every $p\in\mathbb{R}^n$ the map \begin{align} \Phi_p\colon\mathbb{R}^n&\to T_p\mathbb{R}^n\\ v&\mapsto D_{v,p} \end{align} is an isomorphism, where \begin{align} D_{v,p}\...
1
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0answers
23 views

Trouble with structure sheaf, derivations, tangent space

I'm teaching myself differential geometry, and the author I've been reading opts to work with germs of locally defined functions on a (not necessarily smooth) manifold to describe the tangent space. ...
2
votes
1answer
44 views

Confusion about the gradient term in the directional derivatives of a vector

Definition: Let $f$ be a differentiable real-valued function on $\mathbb{R}^3$, and let $\mathbf{v}_p$ be a tangent vector to it. Then the following number is the derivative of a function w.r.t. the ...
1
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0answers
14 views

The tangent space $T_pM$ in terms of the gradient

Let $f: {\mathbb R}^n \to \mathbb R$ be a ${\mathbb C}^1$ function. The graph of $f$ is the surface $M :=\{(x, f(x)) \in {\mathbb R}^n \times \mathbb R | x \in {\mathbb R}^n\}$. Given an arbitrary ...

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