# 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 ...
• 123
1 vote
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### 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 ...
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### 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 ...
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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 ...
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1 vote
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### 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 ...
• 202
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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 ...
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• 615
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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 ...
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• 751
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### 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 ...
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### 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. ...
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### 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 ...
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### 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\...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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1 vote
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### 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 ...
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### 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 $Df|_x: E \to E'$. ...
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### 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 ...
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### 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 ...