Questions tagged [co-tangent-space]
Use this tag for questions about the dual space at a point of the tangent space.
66
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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 $p \in M$,
$$g_p: T_pM \times T_pM \rightarrow \...
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Tangent vector and the cotangent space
I tried to prove that $\{dx^i\}$ is the basis of the cotangent space $T_pM$ of a manifold $M$ for $x^1,\dots,x^n$ local coordinates in neighborhood $U$ of $p\in M$.
I reed somewhere that $dx^iX_p=X_p(...
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Understanding a passage describing the state and phase space of a classical particle
I am reading lecture notes on mathematical physics and I have a question regarding the following passage:
In classical mechanics, the state of a particle at time $t$ is uniquely determined by its ...
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Find the cotangent space of $(1,1,1)$ and a system of parameters for $V(z^2-xy,x^3-yz)$
Suppose you have the variety $X=V(z^2-xy,x^3-yz) \subset \mathbb{C}^3$, and that you want to find:
The dimension of $X$;
The cotangent Zariski space of the point $(1,1,1)$;
A system of parameters for ...
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Cotangent space and pushforward
I wonder if there is a connection between the following two notions:
Pushforward: For a smooth map $f:M\to N$ between smooth manifolds $M$ and $N$, we define the pushforward
$$df:TM\to TN$$
between ...
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basis of cotangent space unclear
Let $M$ be a manifold, $T_xM$ its tangent space with basis {$\frac{\partial}{\partial x_i}$}, with $x_i$ being the $i$-th coordinate function of a chart. A cotangent space $T_x^\star X$ is defined as ...
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Non-degeneracy of the Poisson structure.
I am reading a book on Poisson manifold. There I found the notion of non-degenerate Poisson structure.
Definition $:$ Let $M$ be a Poisson manifold with Poisson bivector field $\pi.$ Then $\pi$ ...
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Motivating the standard cotangent bundle Lie group structure
If $G$ is a Lie group with product $\circ: G \times G \to G$, an "obvious" Lie group structure present on the tangent bundle $T G$ is given by taking the differential of the Lie group ...
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Showing that $I/I^2 \simeq T_{x_0}^{\ast} X.$
Let $X$ be a Poisson manifold with Poisson bivector field $\Pi.$ Let $x_{0} \in X$ be such that $\Pi (x_{0}) = 0.$ Let $\mathcal O (X)_{x_{0}}$ denote the ring of germs of the smooth functions on $X$ ...
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Issues with Definitions of Mathematical Objects in General Relativity
I am a physics student and I am currently taking a course on GR, but the module is not presented in a particularly rigorous manner. My problem lies in one line of the online notes (which unfortunately ...
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Are dual vectors always equal to or in the direction of gradients?
Are the gradients of coordinates, like $\operatorname{grad} u$, $\operatorname{grad} v$, in a non-orthogonal coordinate system of a surface like $(u,v,u^2 +3uv)$, still equal to the dual vectors? (I ...
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What type of quantity is absement?
In differential geometry, position is a point in a manifold, velocity is a vector in the tangent bundle, and acceleration is a quantity in the double tangent bundle (or the tangent bundle if a ...
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Cotangent vectors as equivalence classes of co-curves?
In manifold theory, tangent vectors can be seen as equivalence classes of curves. Using the definition in Weintraub's Differential Forms: Theory and Practice, let $M$ be an $n$-dimensional manifold ...
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Differential form in a variety, from definition and properties
Following Shafarevich Reid - Basic algebraic geometry 1, the differential at $p$ of a polynomial $F(T_1,\dots,T_n)$ is the linear part of the Taylor expansion at $x$ (p. 87), so
$$ (dF=)d_xF=\sum \...
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Tangent complex to a cotangent stack
Given a smooth Artin cotangent stack $X = T^{*}Y$ can one describe the two-term tangent complex $T_X$ in terms of $T_Y$?
The example I am interested in here is the stack of Higgs bundles over a curve $...
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basis vector of tangent space of manifold
In this article https://en.wikipedia.org/wiki/Tangent_space, under the section title Basis of the tangent space at a point, it says "... Then for every tangent vector v\in TpM, one has...." ...
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Calculation of the Lie derivative of the fundamental one-form in three different ways
I am a physicist who is trying to understand more formal differential geometry in the context of classical mechanics. I came across three ways of computing the Lie derivative of differential one-forms....
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Why is the phase space expressed in terms of the cotangent space of the configurational space?
The following is taken nearly verbatim from section 1.1.2.1 Free Energy Computations: A Mathematical Perspective by Mathias Rousset, Gabriel Stoltz and Tony Lelievre.
An excerpt, in which the part I ...
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Compute the dimension of the quotient space $\mathfrak m_a/\mathfrak m_a^2$
Let $\Omega$ be an open set of $\mathbb R^n$ and $\mathcal F(\Omega)$ be the unital commutative ring of infinitely differentiable real functions over $\Omega$. Given $a\in\Omega$, consider $\mathfrak ...
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Explicit evaluation of symplectic form on cotangent bundle
Let $M$ be a manifold and denote by $T^*M$ its cotangent bundle.
Let $(x,U)$ be a coordinate chart so that $x: U\to \mathbb{R}^{n}$.
Let $p\in U$ and $v\in T^*_pM$, then we can write
$v = v_i dx^i$
...
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First Chern class of the cotangent bundle vanishes
I'm interested in the first Chern class of the cotangent bundle. I concretely work on the sphere $S^2$, but the reasoning below seems to work for any manifold.
I take the symplectic point of view ...
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Problem with goniometric function and triangles.
I can't resolve this problem, above there is what I've done. Can you help me? Is there a better way to solve this problem?
A triangle $\triangle{ABC}$ is given, where the angle $\angle{BAC}=2{\alpha}$ ...
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References on the ''bi-symplectic'' structure of $T^\ast T \mathcal M$.
Let $\mathcal M$ be a smooth manifold. One can consider two tautological one forms on $T^\ast (T \mathcal M)$:
("Diagonal case") The usual tautological one form on $T^\ast \mathcal E$, ...
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How do we construct the cotangent space?
I am following a graduate course in algebraic geometry and our professor introduced last week the cotangent space at a point p of a variety as the quotient $m/m^2$ where $m$ is the maximal ideal of ...
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Different notation and a little more insight for $dx^i$ based on an alternative definition of the cotangent space
In this post the cotangent space $T_p^* M$ was defined as the quotient $C^\infty(M)/W_p(M)$ where $W_p(M) \subset C^\infty(M)$ is the subspace of smooth functions with vanishing 1st derivative (i.e. ...
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Metric, tangent and contangent spaces
I have a lot of confusion understanding manifolds, charts, tangent and cotangent vector spaces…
Please let me know where I am wrong.
Let’s take a flat minkosky space in 4 dimensions and call it $M$, ...
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Possible definition of the cotangent space
This definition is inspired by the book mentioned in this question. As it turns out later it is identical to that in question mentioned in the comment by Paul Frost.
The tangent space at $p \in M$ can ...
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Contangent space as a jet space, (inconsistency ?), Renteln
In Renteln's, Manifolds, Tensors and Forms, p. 81, The cotangent space as a jet space$^*$, we have the following definitions
Let $f:M \to \mathbb R$ be a smooth function, $p \in M$, and $\{x^i\}$ ...
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Confusion about role of covectors in behavior of momentum
This question has been beaten to death on math stackexchange, so I'm a little embarassed I have to ask again, but I am really confused about how/why momentum is a covector. I read Arnold but that didn'...
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symplectic geometry: help showing the cotangent lift of an action to a symplectic manifold is a symplectic action
I am following da Silva's lectures on symplectic geometry.
She defines the lift of a diffeomorphism as follows:
Let $X_1$ and $X_2$ be $n$-dimensional manifolds with cotangent bundles $M_1=T^*X_1$ and ...
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Why such cotangent space have $dx_i$ instead of $d/dx_i$ as part of their basis?
DEFINTION OF TANGENT BUNDLE:
Given a smooth manifold $M \subset \mathbb{R}^{n}$ embedded as a
hypersurface represented by the vanishing locus of a function $f \in
C^{\infty}\left(\mathbb{R}^{n}\right)...
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isomorphism of (co)tangent space
Let $X$ and $Y$ be Riemann surfaces. $f:X→Y$ be analytic map. Suppose $f$ induces isomorphism $f*$ between (co)tangent spaces, then, $∀p∈X$, $f'(p)≠0$.
How can I prove(understand) this?
P.S. Such $f$ ...
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Specific example Pull-back equality in manifolds
I have to prove the next equality
$$
\psi^*\left(\frac{xdx+ydy+zdz}{x^2+y^2+z^2}\right)=dt
$$
wher $\psi^*$ is the pull-back of
$$
\psi:S^2\times\mathbb{R}\rightarrow \mathbb{R}^3-\{(0,0,0)\}, \
$$
$$
...
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Cotangent space as a quotient
Let $M$ be a smooth manifold, $p \in M$. Let $I_p$ be the subspace of $C^{\infty}\left ( M \right )$ consisting of smooth functions that vanish at $p$, and let $I_p^2$ be the product of $I_p$ with ...
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Relation between different definitions of the cotangent space of a smooth manifold
There are two ways known to me to define the cotangent bundle on a smooth manifold $X$:
Either as the dual bundle of the tangent bundle (see any textbook on differential geometry) or (by abuse of ...
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1-forms vs k-forms vs Vector Fields
I'm trying to get a comprehensive understanding of similarities and differences of differential forms and vector fields. As I understand, 1-forms are analogous to vector fields in that for a vector ...
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Confused regarding the (dual) basis vectors of the cotangent space $T^*_pM$ of a smooth manifold $M$ and components.
Let $M$ be a smooth $n$-manifold with $p\in M$. Consider the cotangent space $$\operatorname{Alt}^{1}T_pM = \operatorname{Hom}(T_pM,\mathbb{R}) = T^*_pM$$
Now, choosing a chart $(U,\varphi)$ around $p$...
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Pullback of a Volume Form
Suppose we have a a diffeomorphism between two manifolds, $f: M \rightarrow N$ and a volume form $\Omega$ on $N$. Then is it true that $f^{*}(\Omega) = \Upsilon$ will always be a volume form on $M$?
...
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Second derivatives, Hamilton and tangent bundle of tangent bundle TTM
I'm learning the Hamilton formalism of classical mechanics, where a second order differential equation is formalized as two first order differential equations on the cotangent bundle of the ...
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Tautological 1-form on the cotangent bundle is intrinsic using transformation properties
I'm following this lecture in symplectic geometry and I'm trying to show the result stated at 31 minutes that the canonical 1-form on the cotangent bundle $M = T^*X$ is well defined regardless of ...
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Definition of Wave map on manifolds
Let $u: V \rightarrow M$, where $(V,g)$ is a Lorentzian manifold and $(M,h)$ is a Riemannian manifold.
The wave equation is defined as $g. \nabla^2 u$.
As far as I see, $\nabla^2 u \in \Gamma(T^{*}V \...
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Reason for defining $df(v)=D_vf$
I'm reading "A Visual Introduction to Differential Forms and Calculus on Manifolds", specifically the section on equivalence of directional derivatives to vectors acting as operators on functions ...
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Attempt to give an alternate definition of Cotangent Space in terms of integration
Let $M$ be the manifold, then we know that tangents space has basis $\frac{\partial}{\partial{x_i}}$ and it acts on $g:M \rightarrow \mathbb{R}$ as $\frac{\partial{g}}{\partial{x_i}}$
and cotangent ...
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local to global definition of symplectic form on cotangent bundle
Show that the form $\omega$ defined locally as $$\omega = \sum dx_i \wedge d\xi_i$$ is globally well-defined on $T^*M$ and restricted to the zero section of $T^*M$ vanishes.
Here we consider $M$ to be ...
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$\mathscr{C}^{k-1}$-manifold structure on the cotangent bundle
Let $X$ be a $\mathscr{C}^k$-manifold of dimension $n$ and atlas $(U_\alpha,\varphi_\alpha)$ (i.e. its atlas is such that the transitions maps $\varphi_\beta\circ\varphi_\alpha^{-1}$ are $\mathscr{C}^...
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What are differential forms?
For a manifold $M$, if we want to speak of "tangent vectors," we often say the tangent bundle $TM$ is the space of tangent vectors. This is sort of an abuse of terminology, I guess you could say, ...
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Struck in Diffeomorphism invariance of the line integral
I have been trying to study The Cotangent bundle from An introduction to smooth manifolds by John M Lee. I have been struck at a specific point in the Diffeomorphism Invariance of the integral. The ...
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Higgs bundle and spectral curve
I have trouble understanding the following part in E. Wittens paper "More On Gauge Theory And Geometric Langlands":
The context is the following: $(E, \varphi)$ is a Higgs bundle with structure group ...
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Canonical bundle definition
In the article 'More on Gauge Theory and Geometric landlands' by Edward Witten, I have read the term of a 'Canonical bundle'. I've looked that up and as far as I understand, the definition is :
$M$ ...
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Question concerning cotangent spaces of Riemann Surfaces
I have some questions about definitions from Otto Forster's Lectures on Riemann Surfaces.
The setup: Let $X$ be a Riemann surface, and allow $U \subset X$ to be open. Define $\mathcal{E}(U)$ to be ...
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