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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How do I prove $F_*Z=(Z^i\circ F^{-1})\partial_i'$, where Z is a field, and $(F(U),x\circ F^{-1})$ a chart with coordinate fields $\partial_i'$?

I am teaching myself differential geometry on manifolds with some notes a professor gave me. As an initial calculation to prove that the Levi Civita connection is invariant under isometries, the ...
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The push forward $\gamma_{*}(\partial t)$ of $\partial t \in T_{t_0}\mathbb{R}$ is the tangent vector $\gamma'(t_0)$

I am trying to understand the following: Let $\gamma: \mathbb{R} \supset I \to M$ be a smooth curve (M is a smooth manifold). Then the push forward $\gamma_{*}(\partial t)$ of $\partial t \in T_{t_0}\...
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The only maximal ideal of the set of all function germs around $p$

Here is the definition we are using for the set of all function germs around p: Now, I want to show that $m(p) := \{\bar{\phi} \in \mathcal{\varepsilon}(p)| \bar{\phi}(p) = 0\}$ is the only maximal ...
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Showing that the vector space structures induced by $\alpha$ and $\beta \alpha$ are equal(#3.3.13).

In the context of "The Tangent Space" and after defining "Germs" and to prove that the vector space structure on the tangent space does not depend on the choice of charts, here is ...
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2 votes
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Is the tangent plane to the surface $x^4+y^4-z^2=0$ at $(0,0,0)$ undefined?

In calculus, the tangent plane to a surface $f(x,y,z)=0$ at $(x_0,y_0,z_0)$ is defined as the plane passing through $(x_0,y_0,z_0)$ with the normal vector $\mathrm{grad} f(x_0,y_0,z_0)$. According to ...
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Tangent space of $S_1$

Let $(x,y) \in S^1$. Then we can calculate $$T_{(x,y)} S^1 = \{(a,b) \in \mathbb{R}^2 : ax + by = 0\} = span\{(y,-x)\}$$ But I read somewhere the tangent space is spanned by $y \partial_x - x \...
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1 answer
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Bound on dimension of tangent space of an affine variety [duplicate]

I've been reading through my notes and the following fact is stated without any proof or justification: For an affine variety $X\subset\mathbb{A}^n$ and a point $p\in X$ we have $$dim_k T_pX\geq dim_p ...
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  • 201
1 vote
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Subsets of a torus that are attainable from a given point by a given distribution (in the differentio-geometric sense)

Given is a $M$-dimensional torus and a known $M'$-dimensional involutive distribution $\triangle$, $M' < M$, on this torus. (Furthermore, $M'$ is known as a function of $M$.) If ${\bf x}$ is a ...
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1 vote
1 answer
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Determine the Lie algebra of the unit-quaternions $S^3 \subset \mathbb{H}^*$

Determine the Lie algebra of the unit-quaternions $S^3 \subset \mathbb{H}^*$ and their left-invariant vector fields. Unfortunately I am struggling with quaternions. I computed the differential of left-...
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3 votes
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Doubt in identifications for a vector field in the product of manifolds

The following is the Exercise 1) a), Chapter 6 from do Carmo, Riemannian Geometry: Let $M_1$ and $M_2$ be Riemannian manifolds and consider $M_1 \times M_2$ with the product metric. Let $\nabla^1$ ...
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1 answer
52 views

What is the tangent space of a matrix subgroup at $A$

I know that the Lie algebra $\mathfrak{g}$ of a matrix subgroup $G$ is its tangent space at $I$, but what would its tangent space be at $A \in G$? If I fix $B_\epsilon = \{A \in M_n \mid \|A\|_2 \le \...
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1 vote
1 answer
93 views

Clarifying notation of derivative in differential geometry

Let $M$ be a smooth manifold and $f: I \to M$ be differentiable, where $I \subseteq \mathbb{R}$ denotes an interval. I have seen the notation $\frac{d}{dt}$ used in this context, for example $\left.\...
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Vector Field Definition(on open subsets of $\mathbb{R}^n$).

I'm reading An Introduction to Differentiable Manifolds and Riemannian Geometry, Revised by Boothby. I'm reading the second chapter right now, and in this chapter, the tangent space $T_p(\mathbb{R}^n)$...
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  • 1,352
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1 answer
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Are partial derivatives of parametric surfaces always orthogonal?

For a surface $\mathbf{r} = \mathbf{r}(s, t)$ are the partial derivatives $\mathbf{r}_s$ and $\mathbf{r}_t$ in general orthogonal? I was thinking the surface of a sphere and in that case indeed these ...
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3 votes
1 answer
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Alternative concepts for tangent spaces of smooth manifolds and derivatives of smooth maps

The derivative of a smooth map $f : U \to V$ between open subsets $U \subset \mathbb R^m, V \subset \mathbb R^n$ at $p \in U$ is a linear map $df_p : \mathbb R^m \to \mathbb R^n$ which is ...
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2 answers
57 views

Finding the inverse of the differential $d\pi_{i_p} : T_p( M_1 \times \dots \times M_k) \to T_{p_i}M_i$

Suppose that $M_1, \dots, M_k$ are smooth manifolds. Show that for each $i$ the projection $\pi_i : M_1 \times \dots \times M_k \to M_i$ is a smooth submersion. I've shown the smoothness of the ...
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How to understand the intrinsic definition fo the projective tangent space

Introduction I am a physicist struggling with some basic definitions and concepts from algebraic geometry. Therefore I apologize if I make mistakes, I'm just learning! By the way, English is not my ...
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How to understand the geometric concept of tangent variety?

Introduction I am a physicist struggling with some basic definitions and concepts from algebraic geometry. Therefore I apologize if I make mistakes, I'm just learning! By the way, English is not my ...
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1 vote
0 answers
87 views

What is a k-tangent variety?

Introduction I am a physicist struggling with some basic definitions and concepts from algebraic geometry. Therefore I apologize if I make mistakes, I'm just learning! By the way, English is not my ...
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2 votes
0 answers
32 views

Comparing the dimension of an algebraic variety and its tangent space [duplicate]

If $X\subseteq\mathbb{A}^N$ is a variety of dimension $n$, I would like to prove that $$\dim T_xX\geq\dim X=n$$ for any point $x\in X$. My first attempt was to use the Krull's Principal Ideal Theorem, ...
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2 votes
1 answer
52 views

Proving that the tangent space, considered as a set of equivalence classes of contours, is a vector space.

[Note: Before you simply paste a link, I've already read several other similarly-phrased posts on this site, none of which quite answer my question or phrase it in a way I understand.] Wikipedia ...
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0 votes
1 answer
76 views

Tangent Space Projector from Riemannian Metric

Consider a $d$-dimensional Riemannian manifold embedded in Euclidean space $\mathcal{M}\subset \mathbb{R}^N$ endowed with a metric $g$. We are given the Riemannian metric tensor $g_{ij}$ for this ...
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2 votes
0 answers
91 views

Motivation for Proposition 3.14 from Lee's Introduction to smooth manifolds

In Lee's book Smooth manifolds he introduces the following proposition Proposition 3.14 (The Tangent Space to a Product Manifold). Let $M_1,\dots, M_k$ be smooth manifolds, and for each $j$, let $\...
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1 answer
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Problem with understanding how abstract tangent space basis vectors are conneted to e.g. polar coordinates

How does the abstract basis of the tangent space $\frac{\partial}{\partial x^i} f := \partial_i (f \circ x^{-1} ) (x(p))$ at a Point $p \in M$ where a function $f$ is empolyed are related to the well-...
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1 vote
2 answers
59 views

Elements of the tangent space $T\mathbb{S}^n$ are orthogonal to a radial point $p \in \mathbb{S}^n$ [duplicate]

I'm asking for some proper source, proof or a counter example for the following thought: Let $\mathbb{S}^n$ be the $n$ dimensional sphere. It seems quite obvious from the cases $n = 1$ (circle) and $n ...
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Partial derivatives $\frac{\partial}{\partial x^1},\dots,\frac{\partial}{\partial x^n}$ form a smooth frame on $\mathbb{R}^n$

Tu defines (in An Introduction to Manifolds, pp. 137-138) smooth frames as a collection of smooth sections $s_1,\dots,s_n$, s.t. $s_1(p),\dots,s_n(p)$ form a basis for the fiber $\pi^{-1}(p)$ in the ...
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1 vote
0 answers
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Tangent space of a Vector space is isomorphic to the vector space

I am trying to prove this statement from John M. Lee's book on smooth manifolds: For a normed vector space $V$, the map $v\mapsto D_{v|a}$ such that $D_{v|a}(f)=\frac{d}{dt}\bigg|_{t=0}f(a+vt)$ is an ...
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2 votes
1 answer
74 views

How to get from a tangent space to the underlying manifold?

In a Lie group like $SO(3)$, it is possible to create a tangent space $\mathcal{T}_pSO(3)$ at a point $p \in SO(3)$. The tangent space has its basis vectors, which span a local linear coordinate ...
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0 answers
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On directional derivatives as tangent vectors and derivations [duplicate]

In differential geometry I'm told that directional derivatives can be interpreted as tangent vectors and I'm trying to build some intuition for this using simple cases. If I take a map $f: \Bbb R^2 \...
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3 votes
0 answers
43 views

Why is it possible to derive the Lie algebra given its Lie group's constraint?

I would like to intuitively understand how to derive the Lie algebra (the tangent space at the identity element $\mathbf{I}$?) of a Lie group $\mathcal{G}$ given the constraints that apply for the ...
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  • 165
1 vote
1 answer
51 views

Zariski tangent space and exactness of $\operatorname{Der}_R(A,-)$ functor

Let $A$ be an $R$-algebra (for $R$ a commutative ring). Let $\def\Der{\operatorname{Der}}\Der_R(A,-): A-\mathrm{mod}\to A-\mathrm{mod}$ be the covariant functor, where $\Der_R(A,M)$ is the set of all $...
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1 vote
0 answers
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Examples of Covariant Differentiation

I am trying to gain a better feel for Covariant Differentiation. I am thinking of the circle , parametrized as $ C(t):=(\cos t, \sin t) ; 0\leq t \leq 2\pi)$. Is it correct to say that it's derivative ...
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0 answers
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A question about the tangent bundle of the tangent bundle

Let $M$ be a smooth n-dimensional manifold, then the tangent bundle $T(M)$ is a manifold which has a tangent bundle itself which we will indicate as $T(T(M)).$ Let $\pi$ be the canonical projection, ...
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2 votes
0 answers
104 views

Equivalence between two definitions of Complex Tangent Space

I have two definitions of "Complex Tangent Space" over a Complex Manifold $M$ of (complex) dimension $n$. One of them is defining, over the real tangent space $T_p(M)$, the complex structure ...
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1 vote
1 answer
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Let $\pi$ be the quotient map $S^n\to\mathbb R\mathrm{P}^n$. Is $\pi_{\star p}:T_pS^n\to T_{\pi(p)}\mathbb R\mathrm P^n$ an isomorphism?

I am trying to prove the following statement: Let $\pi$ be the quotient map $S^n\to\mathbb R\mathrm{P}^n$. Then $\pi_{\star p}: T_pS^n\to T_{\pi(p)}\mathbb R\mathrm P^n$ an isomorphism Here is my ...
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  • 493
1 vote
0 answers
93 views

What is the derivative of the inclusion map $\iota: M \rightarrow G \times_H M$

Let $G$ be a compact Lie group and $H$ be a Lie subgroup of $G$. Suppose that $M$ is a smooth manifold on which $H$ acts from the left. Let's consider the action of $H$ on $G \times M$ : $$h((g,m)):=...
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2 votes
0 answers
183 views

Tangent space of $G \times_H M$

Let $G$ be a Lie group and let $H$ be a Lie subgroup of $G$. Let $M$ be a smooth manifold on which $H$ acts from the left. Let's consider the action of $H$ on $G \times M$ : $$h((g,m)):= (gh,h^{-1}m)...
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0 votes
1 answer
37 views

When defining tangent spaces, given $\gamma:(-1,1)\to M$ and $\varphi:U\to\mathbb R^n,U\subseteq M$, why is $\varphi\circ\gamma:(-1,1)\to\mathbb R^n$?

So I was reading an article on Wikipedia on tangent spaces but I have a question. So the definition is as follows, Suppose that  $M$ is a $C^{k}$ differentiable manifold (with smoothness $k\ge 1$ and ...
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2 votes
1 answer
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The derivative of a smooth map between smooth submanifolds of Euclidean spaces: Is it a best linear approximation?

Let $M \subset \mathbb R^m$ and $N \subset \mathbb R^n$ be smooth submanifolds and $f : M \to N$ be a smooth map. We can define the (Euclidean) tangent space $\tilde T_pM$ at $p \in M$ as the set of ...
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1 vote
0 answers
33 views

Tangent space of $G/H$ at the identity

Let $G$ be a compact Lie group and let $H$ be Lie subgroup of $G$ with Lie algebras $\mathfrak{g}$ and $\mathfrak{h}$ respectively. 1)- How do we prove that : $$T_{[e]} (G/H) \simeq \mathfrak{g}/\...
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5 votes
2 answers
50 views

Integral curve definition in terms of the tangent vector: why $D \varphi_{t} (d/dt) = X_{\varphi(t)}$?

I have this definition of an integral curve from these notes, page 29, on differentiable manifolds. With a manifold $M$, an integral curve of a vector field $X$ is a smooth map $\varphi: (\alpha,\beta)...
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1 vote
0 answers
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How to interpret the tangent vectors in easy situation?

In differential geometry, given a manifold with coordinates $q^i$, the tangent basis vectors are defined as $(\frac \partial {\partial q^i})$. This definition is very general so, even if it not ...
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1 vote
1 answer
34 views

Existence of specific curves necessary to construct $T_p(M)$

Let $M$ be a manifold of dimension $d$, and $p$ some fixed point on $M$. Define the tangent space $$T_p(M):=\{v_{\gamma,p}:\gamma\text{ is a curve in $M$ passing through $p$}\}.$$ We want to construct ...
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2 votes
0 answers
42 views

Do equivalence classes of mutually tangential curves depend on choices of charts from inequivalent atlases?

Considering a topological manifold $\mathcal S$, let's say specificly of ${\rm dim} = 2$ (i.e. "a surface") we can also identify curves (the set of curves $\{ \mathcal K_j \}$) "in"...
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  • 459
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0 answers
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Showing a linear map is a derivation at a point iff it vanishes on the rank 0 map

I am again having a hard time solving an "easy" exercise of the lecture note by Ed Segal on the theory of manifold. (http://www.homepages.ucl.ac.uk/~ucaheps/papers/Manifolds%202016.pdf) The ...
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2 votes
1 answer
74 views

Tangential planes on a surface at the points of intersection with the sphere

I was looking at a question in an old exam at it happens that someone has already asked it. However, I have one question regarding the formulation of the answer: Prove that tangent planes to the ...
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2 votes
1 answer
61 views

Tangent vectors of smooth manifolds as "modified" derivations?

It is a well-known approach to define the tangent space $T_pM$ of a smooth manifold $M$ at $p \in M $ as the vector space of derivations $d : C^\infty(M) \to \mathbb R$ at $p$. Here $C^\infty(M)$ ...
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1 vote
0 answers
30 views

Tangent space to a level curve

Let $\Omega$ be an open set of $\mathbb{R}^n$. Let $f:\Omega\to \mathbb{R}$ a differentiable function, and $X=\{ x\in \Omega \, | \, f(x)=c\}$ a level curve. If $v$ is a tangent vector to $X$ in $a$, ...
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2 votes
1 answer
70 views

Change of basis in tangent space of $C\equiv x^2+y^2=z^2, \, z>0$

Let $C$ be the smooth manifold defined by $x^2+y^2=z^2$ for $z>0$ and consider the following parametrizations: $$\begin{array}{rcll} \Phi:&\mathbb R^+\times (0,2\pi)&\longrightarrow &C\\...
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  • 493
1 vote
0 answers
62 views

Why is $\{p\}\times W$ is a subspace of $T_pS$, and equal to it (S an affine subspace of $\mathbb{R}^n$)?

I am having trouble with the following example of tangent space. I guess the book is treating tangent vectors as tangent to curves for this part, so I am trying to make sense of it accordingly Tangent ...
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