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Questions tagged [tangent-bundle]

The tangent $TX$ of a smooth (real or complex) manifold is defined as disjoint union of all the tangent space at the points of $X$. This the first and natural example of vector bundle.

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Computation of the push forward of vectors

I am trying to understand the push forward of a vector field by going through a specific calculation. Consider $f: \mathbb{R}^3 \rightarrow \mathbb{R}^2$ given by $f(x,y,z) = (x+y+7, z-x-5)$ and ...
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Is my understanding about the smooth structure on tangent bundle accurate?

I'm not sure if I understood the smooth structure on tangent bundle of a smooth manifold, so my question is whether my understanding is correct, or not, so let me explain what I've understood so far. ...
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For any k-dim. subspace $ L$ of $T_p M$, can we find a sub manifold, say $R$, of $M$ containing $p$ s.t $T_p R = L$

Let $M$ be an $n$ dimensional manifold, and $S\subseteq M$ be a k-dim. sub manifold of $M$, where each is in fact a smooth manifold to be precise. We know that $T_p S$ is a k-dim. subspace of $T_p M$....
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Tangent Map of an Isometry and the Shape Operator

I have spent a lot of time on this problem and would appreciate some help. Please bear with me. Let $M$ be a surface and $F\colon\mathbb{R}^3\rightarrow\mathbb{R}^3$ an isometry. Denote with $F_\...
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Determining the linear independence of tangent vectors at a point on the manifold

We define the tangent space at a point, say $x_0$, on the manifold $M$ as the set of all derivations, i.e maps which maps smooth maps from a neighbourhood of $x_0$ to real numbers to real numbers. ...
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The definition of normal bundle as a quotient.

Let $f: N \to M$ be a smooth immersion and let $p \in M$, $W = f(V) \subset M$ be an submanifold with $q = f(p).$ Then the sequence is split exact $$T_qW \hookrightarrow T_qM \stackrel{\mu}\to T_qM/...
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parallel transport is independent of the bases chosen in each of the two tangent spaces

Connections on principal fibre bundles In the above set of notes on page-3 section 2.2 under the heading Parallel transport there is a statement that equivariance ensures that the parallel transport ...
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The differential $df:T(M)\rightarrow\mathbb{R}$ and the differential map $df:T(M)\rightarrow T(\mathbb{R^1})$ differ by a canonical isomorphism.

This is a problem from Semi-Riemannian Geometry by Barrett O'Neill. For a smooth manifold $M$, we denote by $\mathcal{F}(M)$ the set of all smooth functions $f:M\rightarrow \mathbb{R}$, and by $T(M)$ ...
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Tangent bundle of a trivial bundle

I was asking myself if the tangent bundle of a trivial bundle $\mathcal{P}=M\times V$ with fiber $\pi: \mathcal{P}\to M$ (actually it would be a principal trivial bundle, but I think it doesn't matter ...
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The image of a vector field under the differential of a diffeomorphism is a vector field

I'm learning about vector fields on manifolds, and I'm slightly confused about the following result. Let $M$ and $N$ be differentiable manifolds, $\varphi:M\to N$ a diffeomorphism, and $\mathrm d\...
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Jacobi Fields are geodesics on tangent bundle

Let $(M, g)$ be a Riemannian manifold. Wikipedia states that "Jacobi fields correspond to the geodesics on the tangent bundle". I'm trying to undrestand this statement. Curves $c : I \to TM$ ...
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Completeness of the tangent bundle of riemannian manifold

Let $(M,g)$ be a riemannian manifold and $TM$ its tangent bundle. There are natural riemannian metrics that we can endow the tangent bundle with (for instance the Sasaki metric) and I wonder if for ...
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Change of coordinates and effects on the tangent bundle of a Manifold

I am doing ex. 2.3(2) from Frankel's book "The geometry of Physics". He says to consider the tangent bundle to a manifold M and show: i) that under a change of coordinated in M, $\partial / \partial ...
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Orientation of a Manifold with Trivial Tangential Bundle

Let $M$ be a smooth (eg $C^{\infty}$) manifold. Let assume that $M$ has trivial, oriented tangent bundle $TM$, so $TM \cong M \times \mathbb{R}^n$ for appropriate $n$ and orientable. How to conclude ...
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Algebraic Groups, dual numbers and differentials

I was looking for a method to compute the explicit differential of a regular map between algebraic groups. More precisely if $X$ is a sub-variety in an algebraic group $G$ (say over a finite field $k$)...
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Notation: gradient as vector field

Consider the tangent space $T_p\mathbb{R}^n$, and suppose $\{\big(\frac{\partial }{\partial x^i}\big)_p\}$ is a basis. So my textbook says that the gradient of a function $f$, $f\in C^\infty(U)$, $U\...
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Geodesics in $\mathbb{R}^n$ with the trivial connection

Define geodesic as follows: Given a tangent bundle $TM\rightarrow M$ with connection $\nabla$, a geodesic is a curve $\gamma:I\rightarrow M$ such that $(\gamma^*\nabla)\dot \gamma = 0$. (Notice ...
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Example of a parallelizable smooth manifold which is not a Lie Group

All the examples I know of manifolds which are parallelizable are Lie Groups. Can anyone point out an easy example of a parallelizable smooth manifold which is not a Lie Group? Are there conditions on ...
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If $TM$ is trivial, then $\Lambda^n(M)$ is also trivial and $M$ is orientable

Suppose that $M$ is a smooth $n-$manifold. Suppose $$ TM=\coprod_{p\in M}T_pM $$ is the tangent bundle of $M$. And let $$ \Lambda^n(M)=\coprod_{p\in M}\Lambda^n(T_pM) $$ where $\Lambda^n(T_pM)$ is ...
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Are these definitions of a differential form equivalent?

The definition from my notes says that a differential $k$-form is a section of $\bigwedge^k T^*X \rightarrow X$, so $\omega \in \Omega^k(X)$ would be a map $\omega : X \rightarrow \bigwedge^k T^*X$ ...
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The complex structure on $T\mathbb C^n$

So the real tangent space of $\mathbb C^n$ at the point $p$ is given by $T_p\mathbb C^n=Span_\mathbb R\{\partial/\partial x_1,\partial/\partial y_1, \dots \partial/\partial x_n,\partial/\partial y_n\}$...
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Showing product of two transversal maps is a transversal

Let $X_1,X_2,M$ be finite dimensional manifolds. Consider maps $f_i:X_i\to M$ such that $f_1,f_2$ are transversal with respect to each other, i.e, for $x_1\in X_1, \ x_2\in Y_1$ with $f_1(x_1)=f_2(x_2)...
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Interesting Consequences of Algebraic Tangent Space For $C^k$ Manifolds

We can identify the tangent space to a $C^\infty$ manifold $M$ at a point $p$ as the dual space $(I/I^2)^*$, where $I$ is the ideal of $C^\infty(M)$ consisting of functions vanishing at $p$, then this ...
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The Grassmannian has a non-orientable tangent bundle (in a certain sense)

Let $X$ be a smooth scheme of dimension $r$. Given a rank $r$ vector bundle $\pi: E\to X$, we say that $E$ is orientable if there is a line bundle $L$ on $X$ with an isomorphism $L^{\otimes 2} \cong \...
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Smooth structure and vector bundle structure on $L_{alt}^k(TM)$

I want to exhibit smooth structure and vector bundle structure on $L_{alt}^k(TM)=\bigcup_{p\in M} L_{alt}^k(T_pM)$ where $M$ is a manifold of dimension $n$ and $L_{alt}^k(T_pM)$ is the set of all $k$-...
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$dL_g(1)$ is a section of a tangent bundle

Given a Lie Group, we have that the left multiplication $$L_g:G \to G$$ such that$L_g(x)=g x$ induces the map $$dL_g(1): T_1G \to T_gG$$ If $v \in T_1G$, I have to prove that $dL_g(1)(v)$ is a ...
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A topological problem in defining the tangent bundle

In one of the questions of my homework, tangent bundles are defined as in the picture. So I was wondering how to prove the openness of $\pi^{-1}(U)$, since the question only states that $\pi^{-1}(U)$ ...
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Diffeomorphism between $TM$ and $M\times R^n$

Let $(M,\mathcal{A})$ be a manifold with smooth structure $\mathcal{A}$. For any point $x\in M$, we define a tangent at x by the triplet $(c,x,h)$, where $c=(U,\phi)$ is a chart at $x$, $h\in R^n$ ($n$...
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Zariski Tangent space to the circle $x^2 + y^2 = 5$ at the point $(2,1)$

As an exammple of the scheme, let's try the circle of radius $\sqrt{5}$ which is the equation $X = \{ x^2 + y^2 = 5 \}$. This circle has four integer points $(\pm 2, \pm 1)$ and $(\pm 1, \pm 2)$. As ...
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Given a 2-plane distribution D in $ M^3$ ( 3-manifold, Contact Structure) , find a 1-form Generating D.

I am trying to show a claim that , given a smooth 2-plane distribution D (a subbundle of the tangent bundle of) of a 3-manifold $M^3$, there is a 1-form $w$ generating the plane distribution locally, ...
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How to show that vector field is continuous?

In the book the definition of a a vector field over $U$(open)$ \subseteq S^n$ is given by a continuous map $s: U \to T(U)$ such that $p_U \circ s=id_U$ where $p_U$ is the base point projection from $T(...
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Topology and smooth structure on tangent bundle

My lecture notes on differential geometry read the following (without proof): For $M$ a manifold, let $TM = \bigcup_{p \in M} T_p M$ be the (disjoint) union of all its tangent spaces. Then, there ...
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An Orthonormal Frame $(X_i)_i$ which satisfies $\bigtriangledown _{X_i} X_j =0$ at a point

I am an undergrad student learning Riemannian geometry. My question is about whether you have a nice orthonormal frame in the following sence. Let $(M, g)$ be a Riemannian manifold, with $\...
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1answer
48 views

Exponential map, Manifold

For each $x\in M,X\in T_x M$ there exists a maximal interval $I_X\subseteq \mathbb{R}$ with $0\in I_X$ and a geodesic $c=c_X$ $$c:I\to M$$ with $c(0)=x,\dot{c}(0)=X$. Let $$C:=\{X\in TM | 1\in I_X\}$$ ...
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Why is a connection on the bundle $SO(M)$ metric compatible?

If we have an orientable manifold $M$ with a metric $g$ and signature $(r, s)$, we can define the principal-$SO(r, s)$ bundle $SO(M)$, the bundle of orthonormal frames of $TM$. This is a subset of the ...
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Tangent Bundle topolgy

I am reading Lee's ''Introduction to Smooth Manifolds'' and it states that the tangent bundle TM has a natural topology and smooth structure that makes it a 2n-dimensional manifold. I get most of the ...
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Conormal bundle and lagrangian submanifold

Let $Q_1^{n_1},Q_2^{n_2}$ be smooth manifolds, $\phi:Q_1\to Q_2$ a smooth map and: $$R_\phi:=\{(x, \xi, y,\eta)\mid y=\phi(x), \xi=(d\phi)^*\eta\}\subset T^*Q_1\times T^*Q_2$$ $$\text{graph}(\phi)=...
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Tangent Space to the Quadratic Form

Let $S:=\{x \in \mathbb{R}^n : x^\top Ax = 1 \}$. We know that $S$ is a $(n-1)$-dimensional submanifold of $\mathbb{R}^n$ because it is a regular level set of a smooth function. What is the tangent ...
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Finding the tangent bundle of the two dimensional torus $\mathbb{T}^2$

I didn't understand clearly the notion of tangent space. For example, I want to find the tangent space in a point of the two dimensional torus $\mathbb{T}^2 = S^1 \times S^1$ or $\mathbb{T}^2 = \...
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1answer
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Continuity of Riemannian norm

I just started learning something about riemannian manifolds. I was wondering if the norm, as a map between the tangent bundle and $\mathbb{R}$ is continuous. Is it? Thanks
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Vector fields on $S^{4n-1}$ spheres

I need to prove that there exists 3 independent vector fields on spheres of dimension 4n-1. I decided to try induction on n. The case $S^{3}$ was covered in 2 or 3 chapters back but i have no idea or ...
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On the $\pi$-induced pushforward of a tensor field on $TM$

A short informal premise. As far as I understand, given a smooth map between manifolds $f: M \to N$ and a smooth vector field $X: M \to TM$, the pushforward $f_* X$ only defines in general a vector ...
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1answer
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Show that differential map $df : TM \to \mathbb{R}$ of a smooth map $f: M \to \mathbb{R}$ is smooth.

I'm trying to prove that for any smooth map $f: M \to \mathbb{R}$ on a manifold $M$ to the real numbers, the map $df : TM \to \mathbb{R}$ from the tangent bundle $TM$ to the real numbers given by $df(...
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1answer
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Well-definition of a derivation

I am trying to understanding a proof of a proposition about the tangent space to an open manifold. Let $M$ be a smooth manifold, $\iota:U\rightarrow M$ the inclusion map for some open $U\subset ...
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Fourier transform, tangent and cotangent bundles

I'm familiar with the notion of Fourier transform in the context of $\mathbb{R}^n$ and more generally, locally compact abelian groups. However recently I came across the Fourier transform acting as ...
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Linearity of map between tangent space and derivation space

Let $v_a:=v|_a:=(a,v)\in\mathbb{R}^n_a:=\{a\}\times\mathbb{R}^n$ be a geometric tangent vector and $D_v|_a\in\mathcal{D}(\mathbb{R}^n)$ be a derivation, to be precise, a directional derivative. For $f\...
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Definition of tangential vectors via charts

Hello fellow mathematicians, I am currently learning about differential geometry and I have read the chapter about the tangential space. Now, there is a note following the definition of tangential ...
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95 views

The differential and smooth sections

I am working on a question regarding a smooth section, which seems quite intuitive in a Euclidian space. Let $M$ be a smooth manifold an $c:(0,1) \to M$ a smooth curve. Show, that $\dot c:(0,1) \to ...
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Coordinate-free proof of non-degeneracy of symplectic form on cotangent bundle

It's relatively straightforward to provide a coordinate-free definition of the symplectic form on a cotangent bundle; the usual way to do this is to construct the tautological 1-form $$\lambda(\xi) = \...
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Trivialization of tangent bundle via direct sum

I was trying to prove the following, but got stuck: given a manifold $M$ and its tangent bundle $TM$, there always exists a bundle $E \rightarrow M$ such that the direct sum bundle $TM \oplus E \...