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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1
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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 ...
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Defining the Tangent space to the boundary of a manifold $T_p(\partial S)$
While studying manifolds I am having some problem with definition of manifold with boundary.Let $S$ be a regular $n$-level surface in $\mathbb R^{n+1}$ with boundary defined by $S=f^{-1}(0)\cap (\...
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Product rule for matrix-vector product
Suppose that $x \in L^2([0,1], \mathbb{R}^m)$ is a vector valued function and $A(x)$ is a ($m \times m$)-matrix whose entries are the components of $x$. Then consider a differentiable curve $\gamma: (-...
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2
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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 $F(p)\in N$...
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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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35
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Disjoint union of $\Lambda^k(T_p U)$ as a bundle.
While studying differential geometry of Manifolds,one has surely come across the space of all alternating $k$-tensors on a vector space $V$ (say of finite dimension $n$). We denote it by
$$
\Lambda^k(...
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Tangent space of a quotient of an algebraic variety
A word of warning: I have no background in algebraic geometry, so please excuse my ignorance. References welcome (but please refrain from saying things like "read Hartshorne's Algebraic Geometry, ...
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An algebraic stack as a non-linear analog of a complex of vector spaces
In their paper Derived Quot Schemes Kapranov and Ciocane-Fontanine write in the introduction:
Indeed, an algebraic stack is a nonlinear analog of a complex of vector spaces situated in degrees $[-1,0]...
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Algebraic tangent space of image of a morphism
I'm getting lost trying to figure out what seems to be a simple question. I've looked in Shafarevich and Görtz & Wedhorn but the points are usually dealt with using schemes, which I haven't gotten ...
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Tangent space at a point of product of algebraic varieties
Let $X$ and $Y$ be algebraic varieties over an algebraically closed field $k$. Consider two points $a\in X$ and $b\in X$. I want to prove that the natural projection maps $p_X:X\times Y\to X$ and $p_Y:...
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Formula for the differential between tangent spaces
I am trying to answer the following question:
Show that, for any $h: \mathbb{R}^n \rightarrow \mathbb{R}^k$ smooth and $p \in \mathbb{R}^n$, its differential between the abstract tangent spaces (white ...
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1
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Confusion about the isomorphism between directional derivatives and tangent vectors
A directional derivative $D_v: C^{\infty}(U) \to \mathbb{R}$ is a derivation on $U$ since it is linear and obeys the product rule.
A typical definition of a tangent space at point $p$ on some manifold ...
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1
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Prove tangent space is in a hyperplane
Theorem: Let $A$ be open and $f:A\subset\mathbb{R}^n\to \mathbb{R}$ such that $f$ is differentiable at $a$. Then, the tangent space of $S_{f(a)}$ at $a$ is contained in the hyperplane $\{D_af(x)=0\}$
...
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What does "mod" mean in the context of this tangent space proof in Warner?
I'm trying to read Frank Warner's Foundations of Differentiable Manifolds and Lie Groups and got confused with Theorem $1.17$ as some who does not have a pure mathematics background.
Let $F_m$ be the ...
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Can a Riemannian metric be defined in terms of the cotangent space?
I have always thought of Riemannian metrics as being an inner product assigned to each tangent space. That is, if $M$ is a manifold, then at any point $p \in M$,
$$g_p: T_pM \times T_pM \rightarrow \...
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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 ...
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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 ...
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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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1
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Tangent vector and the cotangent space
I tried to prove that $\{dx^i\}$ is the basis of the cotangent space $T_pM$ of a manifold $M$ for $x^1,\dots,x^n$ local coordinates in neighborhood $U$ of $p\in M$.
I reed somewhere that $dx^iX_p=X_p(...
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1
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Finding the tangent vector of a cubesphere using world coordinates
TLDR.:
I’m trying to find the tangent vector of a cubesphere using only world space coordinates.
Background:
A cubesphere is basically just an inflated cube. It is commonly used to project 2D textures ...
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0
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50
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Consistency between two different ways of defining an orientation on $S^n$
In the book Topology, Geometry and Gauge Fields Interactions by Gregory L. Naber, he defines the standard orientation of $S^n$ (viewed as a subset of $\mathbb{R}^{n+1}$), $n\geq 2$ by the oriented ...
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2
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Singularities of the intersection of quasiprojective schemes
I'm getting hung up on what should be a fairly simple application of definitions in Poonen's paper on Bertini over finite fields. Here's the setup:
$S=\mathbb{F}_q[x_0,\dots,x_n]$
$S_d$ consists of ...
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1
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What properties define the tangent bundle up to diffeomorphism?
In the theory of smooth manifolds there are many ways in which the tangent bundle can be defined, begging the question: what set of properties define the tangent bundle 'up to diffeomorphism'?
These ...
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Book that Develops the Theory of Tangent Space whilst Defining Tangent Vectors as Equivalence Classes of Curves
Currently reading Lee's Introduction to Smooth Manifolds. At the end of chapter $3$ he mentions that tangent vectors may be defined in terms of equivalence classes of curves, but by that time he has ...
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2
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Interpretation of basis of tangent space
Suppose a regular surface $S\subset \Bbb R^3$ parametrized by $\Sigma:U\to \Bbb R^3:(x_1,x_2)\mapsto (x(x_1,x_2),y(x_1,x_2),z(x_1,x_2))$. If $p=\Sigma(u_0,v_0)\in S$, then the tangent space to $S$ at $...
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Tangent vectors as differential operators vs equivalence classes of curve in manifold
Suppose we have a curve $\alpha: \mathbb{R} \rightarrow \mathbb{R}^2$ through $p \in \mathbb{R}^2$ and $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ is a smooth function. Then $f \circ \alpha: \mathbb{R} \...
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On Marsden's 'Introduction to Mechanics and Symmetry' Exercise 4.3-3. (orientability of product manifold)
Given two differentiable manifolds $\mathcal M$ and $\mathcal N$ I needed to show that
$$\mathcal M, \; \mathcal N \mbox{ orientable } \Rightarrow \mathcal M \times \mathcal N \mbox{ orientable.}$$
...
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Is there an Approximate tangent space that is NOT a Classical tangent space?
First given $M \subset \mathbb{R}^n$, and $x \in \mathbb{R}^n$, $r>0$, we define the blow up by:
$$ \Phi_{x,r}(y) = \frac{y-x}{r}.$$
We say $x \in \mathbb{R}^n$ has a $k$ dimensional approximate ...
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Coordinate transformations of the tangent bundle as a manifold.
My question can be summarized as: Let $\mathcal{M}$ be a smooth manifold and $T\mathcal{M}$ be its tangent bundle. It's well known that $T\mathcal{M}$ can be viewed as a smooth manifold. Then how does ...
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tangent vector of a disk embedded in R^3
Consider in $\mathbb{R}^3$ the plane $N: x+y+z=0$ and the sphere $S: (x-a)^2+(y-b)^2+(z-c)=r$, where $r$ is fixed.
The intersection $N\cap S$ is the disk embedded in $\mathbb{R}^3$.
I would like to ...
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Understanding the "abuse of notation" in the differential of tangent vectors
I am reading John Lee's Smooth Manifolds book, current looking at the bottom of Page 63 in which we are working out what the differential looks like in the special case that it's along the transition ...
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0
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47
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The domain of derivative of a function defined on a manifold
For a function $f:X \to R$, where $X$ is a manifold. For any point $x \in X$, the derivative of $f$ at $x$ is $[Df(x)]:T_xX\to R$. And $T_xX$ is the tangent space of $X$ at point $x$. Why is the ...
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1
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Checking smoothness of curves and finding multiplicites.
I am asked to check whether or not this curve is smooth (and if not provide singular points): $x_2^2x_0 = x_1^3 - x_1x_0^2$.
The way I approached this was by to use the projective Jacobi criterion on ...
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Approximate tangent space agrees with tangent space of submanifold of $\mathbb{R}^n$
I am stuck on trying to prove that the approximate tangent space of a submanifold of $\mathbb{R}^n$ agrees with its tangent space. To make things more precise I'll give the relevant definitions. Note ...
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Set of linearly independent vectors in a tangent-space that spans a non-degenerate 2-plane are dense in the cartesian product of the tangent-space.
Let $R$ be the riemann curvature tensor, viewed as a $(0,4)$-tensor field. For $p \in M$, where we have a semi-riemannian manifold $(M,g)$, we can look at linearly independent vectors $u,v \in T_pM$ ...
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How to find the tangent cone of $A = \{(x, y) \in \mathbb{R}^2: \; y = \sqrt{|x|}\}$ at the origin?
My problem is pretty much stated in the title. I don't quite know how to approach this. I have the definition of a tangent cone to a set $K\subset \mathbb{R}^n$ at a point $x$: A tangent vector $l \in ...
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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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Bases for tangent spaces and contractions
Let $M$ be a smooth manifold, $X$ a smooth outward pointing vector field on $\partial M$, and $\omega$ an orientation form on $M$. In Tu's Introduction to Manifolds, he proves that the contraction of $...
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How can I compute the differential of this function between surfaces $C$ and $G$?
Let me consider the surface $S:=\{(x,y,z): x^2+y^2=1\}$ and $G:=\{(0,y,z): y,z\in \Bbb{R}\}$ and define $f:C\rightarrow G$ by $f(x,y,z)=(0,y,z)$. Let us take the following two patches for $C$ and $G$: ...
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The Jacobian of the nearest point map is an orthogonal projection map.
I'm reading a book about harmonic maps and at some point they said something I did not understand. Let $(N, h)$ be a compact Rimannian manifold. By the isometric embedding theorem by Nash, the exists $...
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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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1
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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 ...
- 1,727
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1
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61
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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). ...
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1
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1
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Clarification of the differential in local coordinates
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, ...
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0
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The method of Lagrange Multipliers in Classical Mechanics
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{...
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