# 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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### The tangent space of a ringed space could be a module over a ring?

Quick question : The tangent space of a ringed space could be a module over a ring ? If so, how could it be ? I did not find anything online. Thank you
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### Is the 1-parameter subgroups of noncommutative Von Neumann algebra linked to 1-parameter subgroups of Lie groups?

I have read "Von Neumann Algebra Automorphisms and Time-Thermodynamics Relation in General Covariant Quantum Theories" of C. Rovelli and A. Connes, and I've been asking to myself whether the ...
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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 ...
1 vote
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### Notation for vector tangent to a curve on a differentiable manifold

A result for differentiable maps that I'll use: let $F:M\to N$, where $M,N$ are smooth manifolds and $F$ is a smooth map. If $(U,x)$ is a chart around $p\in M$ and $(V,y)$ is a chart around $F(p)\in N$...
1 vote
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### Pulling back tangent vectors to basis the tangent bundle

Suppose we have a differentiable manifold and coordinate chart $M \xrightarrow{\phi=(x, y)} \mathbb R^2$ Let $f: M \rightarrow \mathbb R$ be a $C^\infty M$ function specified "in $(x, y)$ ...
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### Finding the tangent hyperplane to a unit hypersphere in $n$ dimensions

Please forgive me if this has already been posted, although I could not find any specific question related enough to my problem (or it might be and I just lack the mathematical background to ...
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### How to characterize the tangent space $T_f C^\infty(K, \mathbb{R}^n)$ and paths in $C^\infty(K, \mathbb{R}^n)$

Let $K \subset \mathbb{R}$ be compact. For any function $f \in C^\infty(K, \mathbb{R}^n)$ how would one characterize the tangent space $T_f C^\infty(K, \mathbb{R}^n)$? I am following a set of notes ...
1 vote
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### The relationship of tangent space between submanifold and manifold.

I was confused when I prove the next Proposition appeared in the book Introduction to smooth manifolds by Lee. Suppose $M$ is a smooth manifold with or without boundary ,$S\subset M$ is an immersed ...
1 vote
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### Is the direction derivatives of (orthonormal) normal vector(s) in its own direction in the tangent space?

Let $M$ be a smooth manifold of dimension $d$ embedded in dimension $D>d$. Let $n_1,\dotsc, n_K$ be any orthonormal basis for $N_xM := T_xM^{\perp}$, the orthogonal complement of the tangent space ...
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### For an equation that gives function value at a point, how do I apply a differential operator?

(Note: Using Einstein summation convention throughout) I'm trying to understand Theorem 2.2.1 in Wald's General Relativity book. We have a smooth manifold $M$, with $p\in M$ and $(O,\psi)$ an open ...
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### Difference between tangent spaces of O(n, R) and SO(n, R) [duplicate]

I know that $O(n, \mathbb{R})$ is a group of nxn matrices satisfying $X^TX=I$, and $SO(n, \mathbb{R})$ has additional condition of positive determinant. But I have no idea about how could those groups'...
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### Showing that if two curves are equivalent under a chart then it holds for other every other chart

Fix $p\in M$. Consider all smooth curves $c : (-\epsilon,\epsilon) \rightarrow M$ with $c(0) = p$. We say that two such curves $c_1$ and $c_2$ are equivalent if there exist some smooth chart $(U,\phi)$...
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### Lie algebra of a Lie group (a question from PSE) [duplicate]

I'm going through some notes on group theory for physics. After introducing the concept of Lie group and Lie algebra the writer makes the connection between the two. Let $G$ be a Lie group of ...
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### Try to find a nowhere vanishing vector field $F$ on $S^1$ such that $F(\vec{x})$ is tangent to $S^1$ at $\vec{x}$

The definition of a tangent vector field in my book is the following: This is from "Munkres-Analysis on Manifolds" I want to define a vector field $V$ on a unit circle $S(1)$ that nowhere ...
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### Tangent space at a point in invertible upper triangular matrices

Let $p\in M$ with $M$ being the group of invertible upper triangular matrices. Determine $T_pM$. My idea - I know that $M$ is a submanifold of the vector space $U_n$ (upper triangular matrices). ...
1 vote
Let $f : N \rightarrow M$ be a smooth map. Fix $q \in N$ and consider a curve $\gamma : [0,T] \rightarrow N$ such that $\gamma(0) = q$ and $\dot{\gamma}(0) = v = \frac{d}{dt}\mid_{t=0}\gamma(t)$. Now, ...
The context I am studying constrained systems in Classical Mechanics following the book Anaytical Mechanics. The autor considers a system of $p$ non-holonomic constraints that have the form \begin{...