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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Conormal exact sequence

Let $X$ be a smooth variety and $Y\subset X$ a smooth subvariety with ideal sheaf $\mathcal{I}_{Y/X}$. For $n\geq 2$ is there an analogue of the exact sequence $$0\rightarrow \mathcal{I}_{Y/X}/\...
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Is there a natural connection on $TM$

The Sasaki metric gives a natural way to equip $TM$ with a Riemannian metric in case $M$ is already equipped with a Riemanian metric. Question: Let $M$ be manifold equipped with a connection, is ...
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What do the fibers of the double tangent bundle look like?

Consider the tangent bundle $\pi:TM\to M$ for some smooth manifold. As outlined in the Wikipedia page, we can then consider the double tangent bundle via the projection $\pi_*:TTM \to TM$, with $\pi_*$...
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Tangent (sub)bundle of manifold question

Say we have two manifolds, $M$ and $N$ and let $F:M → N$ be an embedding and $\tilde{M}=F(M)⊂N$. Show that the tangent bundle of M in N, given by $T\tilde{M}=dF(TM)⊂TN|$ $\tilde{M}$ , is a subbundle ...
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What is $\operatorname{Hom}(S,Q^\vee)$ for a Grassmannian?

Let $G(k,V)$ be the Grassmannian of $k$-planes in a complex vector space $V$ of dimension $n$. There is the famous universal exact sequence of vector bundles on $G(k,V)$ $$ 0 \to S \to V \otimes \...
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How to handle the differential of a vector field, ${\rm d}X:TM\to TTM$, in terms of (equivalence classes of) curves?

Given a generic smooth function $f:M\to N$, we know that its differential is a smooth function $\mathrm df:TM\to TN$ such that $$\mathrm df(p,[\gamma'(0)])\equiv (f(p),\underbrace{\big[\partial_t\big|...
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Diffeomorphism between $SO(3)$ and $STS^2$ [duplicate]

I am studying differential geometry. I'm having a problem I don't know, so here is the question. Problem) Show that $SO(3)$ is diffeomorphic to $STS^2=\{v\in TS^2 : \Vert v \Vert_g=1\}$, where $g$ is ...
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Clarification with Pontryagin classes of the 4-dimensional sphere $S^4$

In some homework I am asked the following: what are the Pontryagin classes of the 4 dimensional sphere $S^4$. My doubt is: should I assume that I am actually being asked about the Pontryagin classes ...
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Intuition on the higher order tangent bundles

The first order tangent bundle $TM$ can be thought of as the set of velocities at each point on the manifold. It can be formally defined in one of two ways: it can be the set of derivations on the ...
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Tangent space of a sphere is locally homeomorphic to to $\mathbb{R}^{2n}$

Tangent space of $S^n$ is locally homeomorphic to to $\mathbb{R}^{2n}$. The tangent space is given by $$TS^n=\{(x,v)\in S^n\times R^{n+1} : v\perp x\}\subset S^n\times R^{n+1}$$ I didn't know where ...
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Coordinate basis for the Tangent of the Tangent bundle

I am interested in second order systems on a smooth manifold $M$ and my main reference is from GEOMETRY OF HORIZONTAL BUNDLES AND CONNECTIONS Since I am interested in coordinated basis, fix a chart $(...
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Global section of a smooth fiber bundle is an embedding

Given a smooth fiber bundle E over a manifold $M$, is any global section $s: M \to $ E is an embedding? Here I always mean a locally trivial bundle. I think the above assertion is correct. If so, does ...
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Change of two normal coordinates based on two nearby points?

Let $M$ be a manifold and $L(M)$ be the tangent frame bundle on $M$. Let $\Gamma$ be a linear connection on $L(M)$ which induces a covariant derivative $\nabla$ on $TM$. Let $p, q$ be two ...
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Lemma 2.3. of Riemannian Geometry by do Carmo; Existence/Uniqueness of the Geodesic field $G$ on $TM$

Lemma 2.3 states: There exists a unique vector field $G$ on $TM$ (+) whose trajectories are of the form $t\mapsto (\gamma(t),\gamma'(t))$, where $\gamma$ is a geodesic on $M$. Uniqueness: Suppose ...
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Can we define the Lagrangian vector field on the tangent bundle form variational principle WITHOUT introducing E-L equations on local coordinates?

In Hamiltonian Mechanics on a cotangent bundle $T^*M$, we can define the Hamiltonian vector field by $$ \iota_{X_H} \omega = \mathrm{d}H $$ where $\omega$ is the canonical symplectic 2-form on $T^*M$. ...
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Isomorphisms in tangent bundles

Given the tangent bundle $T$, is the following isomorphism $$\Lambda^{d-1}T\simeq T^*\otimes\Lambda^dT^* $$ true on a $d$-dimensional manifold? i.e. are $(d-1)$-vectors equivalent to objects which are ...
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Tangent bundle at a set

If we have a set $X=A\cup B$ and $Y= A\cap B$. What is the difference between $TY$ (the tangent bundle of $Y$) and $T_{Y}X$ (the tangent bundle of $X$ at $Y$)? That is, $Y$ is subset of $X$, so I can ...
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How is this a tangent vector?

Let $\pi :E\to M$ be a vector bundle with typical fiber $V$. Suppose that $\pi^*E$ is the pullback bundle of $E$ by $\pi$. If $(\zeta, \xi) \in \pi^*E$, then the map $\pi(\zeta +t\xi)$ is constant in $...
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Structure groups of $G$-bundles and $G$-associated bundles

Take a $G$-bundle to be a fiber bundle with typical fiber $F$ whose transition functions take values in the structure group $G$ = Aut($F$). Based on this definition alone, I would assume that the ...
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Deducing exactness of Euler seqeuence from Euler's theorem on homogeneous functions

Question : How to deduce the exactness at $\mathcal{O}_{\mathbb{P}^n}(1)^{n+1}$ of the Euler sequence $0\rightarrow\mathcal{O}_{\mathbb{P}^n}\rightarrow\mathcal{O}_{\mathbb{P}^n}(1)^{n+1}\rightarrow T\...
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Tangent bundle of exterior power

Let $M$ be an $m$-dimensional manifold and consider a map $L \colon \Lambda^{m-1} T^\ast M \to \Lambda^m T^\ast M$. I am trying to look at the derivative of a map of this form, namely $TL \colon T(\...
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Are the horizontal and vertical subbundle preserved under isometry?

Let $(M,g)$ be a Riemannian manifold and $TM$ its tangent bundle, equipped with the Sasaki metric $g_S$. Let $T_1M$ be the unit tangent bundle (consisting of all tangent vectors of unit length), with ...
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Are $\mathbb{R}P^3$ and $T^1S^2$ isometric?

It is well-known that 3-dimensional real projective space $\mathbb{R}P^3$ is diffeomorphic to $T^1S^2$, the unit tangent bundle of the 2-sphere. However, I could not find any reference to whether ...
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Simultaneous rectification of $n$ distributions

Consider a smooth $n$-dimensional manifold $\mathcal{M}$ equipped with local coordinates $x$. In the tangent bundle $T\mathcal{M}$ we consider $n$ rank $1$ distributions $\mathcal{D}_i$ satisfying \...
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How can I compute the velocity vector of a vector field along a curve, considered as path in the tangent bundle?

Consider a Riemannian manifold $M$, a unit speed geodesic $\gamma\colon[0,l]\to M$ and a parallel unit vector field $E$ along $\gamma$. Let $f\colon [0,l]\to \mathbb{R}$ be some smooth function. ...
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Understanding vector field(s) on $\mathbb{S}^3$.

I was slving the exercises of John Lee's book "Introduction to Smooth Manifolds", where there is an exercise asking us to prove that $\mathbb{S}^3$ is parallelizable. In the hint, the author ...
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Expressing complexified tangent bundle of a spin 4-manifold as a Hom bundle

I am reading Moore's book Lectures on Seiberg-Witten Invariants, section 2.2. First here are some defintions that the book uses. The group $\operatorname{Spin}(4)$ is defined to be the product group $...
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Non-compact manifold $M$ has a non-vanishing vector field

I knew this question was asked before, like here Non-vanishing vector fields on non-compact manifolds . But it seems there is no detailed satisfying answer I could find. So can anyone please give ...
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What does "the map $i$ is transverse to the vector field $X$ everywhere" mean?

Geometric Theory of Foliations, Page 28. The Lines 7 and 8 from below. What does "the map $i$ is transverse to the vector field $X$ everywhere" mean?
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Proof verification of the tangent space of the tangent bundle of the sphere using the canonical identifications

Let $\iota : S^m \hookrightarrow \mathbb{R}^{m+1} $ and $\iota_T : TS^m \hookrightarrow TR^{m+1}$ be embeddings and let $J_v : V \xrightarrow{\cong} T_vV$ be the canonical isomorphism where $V$ is a ...
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How to show that the tangent bundle is trivial?

So I was studying for a test and found this practice problem: Let $S^1=\{ (x,y)\in \mathbb{R}^2, x^2+y^2=1 \}$. i) Show that $S^1$ is a smooth manifold in $\mathbb{R}^2$ (Done) ii) Show that the ...
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Tangent bundle on a smooth scheme

My question is, how might I construct the tangent bundle on a smooth scheme? It is clear how to define the tangent space at a point: the Zariski tangent space. It is also clear what we should do in ...
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Doubt in the definition of continuous vector field

I'm wondering how the manifold topology on the tangent bundle $ TM $ yields a notion of continuity between tangent vectors as well, and not merely between base points. Let $X: M \rightarrow TM$ be a ...
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Milnor's notation for tangent bundle

In Milnor's "Characteristics classes" there are two notation for the tangent bundle of a smooth manifold $M$. i.e. $\tau_M$ and $DM$. and $DM_x$ for tangent space. Why he uses of two ...
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Trying to write the bundle structure for $\Lambda^2(T^*(\mathbb P^3\times \mathbb P^3))$

I'm trying to find an atlas and in general the bundle structure for $\Lambda^2(T^*(\mathbb P^3\times \mathbb P^3))$ respect to, for example, the local chart $(U_{01},\psi_{01})$, where $$U_{01}:=\{((x^...
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Taylor expansion of Lagrangian $L = \frac{1}{2} \dot{q} \cdot A(q) \dot{q}-V(q,\dot{q})$

I'd like to understand the Taylor expansion (second order) of the function $L(q,\dot{q}) = \frac{1}{2} \dot{q} \cdot A(q) \dot{q}-V(q,\dot{q})$, where $A$ is the mass matrix and $V$ is of the form $V(...
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Morphism of tangent space induce from a morphism of varieties.

I am trying to show that a morphism of varieties $f: X \to Y$ induces a linear map between the tangent spaces $\tilde{f}: T_aX \to T_{b} Y$, for $a \in X$ and $b = f(a) \in Y$. My idea was to use the ...
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A smooth bundle map from $\wedge^k(TM) \to \wedge^k(TN)$

Suppose $M$ and $N$ are smooth manifolds, and $F:M\to N$ is a smooth map. Does $F$ induces a smooth bundle map from $\wedge^k(TM)$ to $\wedge^k(TN)$, where $TM$ and $TN$ are tangent bundles on $M$ and ...
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Question in a proof from Gathmann's notes on Algebraic Geometry: The tangent space.

I am studying chapter 10 from Gathmann's notes about algebraic geometry and there is something I don't understand in the following proof. I don't really understand the equality $\frac{g}{f} = c g$. ...
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What is the tangent bundle of Mobius band?(It is trivial or not?)

I know that Mobius band is a quotient space of unit square with the equivalence relation $(0,t)\sim (1,1-t)$. I want to find the tangent bundle of Mobius band upto diffeomorphism. I am not getting any ...
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Is there a bundle concept that includes tensors and spinors as special cases?

A scalar field is a map from the base space to the field of interest but it is equivalently a section of a (0,0)-tensor bundle. Similarly, a vector bundle is just a section of a (1,0)-tensor bundle. ...
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Is $TM$ a manifold with or without boundary when $M$ is a manifold with boundary?

I am working on manifolds with boundary. Assume $(E, \pi)$ is a vector bundle over a manifold with boundary $M$, for example $TM$ or $T^*M$. If $\phi : \pi^{-1}(\mathcal{U}) \to \mathcal{U} \times \...
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Second Bianchi identity on tangent bundle

I'm having a hard time on proving the second Bianchi identity in the case of tangent bundle without choosing of metric. I already know that on a general vector bundle, the second Bianchi identity ...
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Two questions about tangent bundles (Resolved)

We can define the tangent bundle in two ways, depending on how we define our tangent spaces. If our tangent spaces are derivations on the space of smooth germs at $p \in \mathcal M$ (where $\mathcal M$...
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Understanding: Differental of Riemannian exponential expressed through Jacobi-Fields

In our DiffGeo-lectures we had a theorem that the differential of the local exponential map can be expressed through Jacobi-Fields: Let $M$ be a Riemannian manifold, $p\in M, X\in T_pM$ such that $exp(...
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Why is the tangent bundle the disjoint union of tangent spaces?

Some books (and actually wikipedia too) define tangent bundle $TM$ of a manifold $M$ as the disjoint union of tangent spaces. Then, they write: \begin{equation} TM=\bigsqcup_{x\in M} T_xM=\bigcup_{x\...
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Killing fields and the Vlasov equation

Let $X$ be a Killing field. Prove that $f\colon (p,V)\mapsto g_p(V,X)$ solve the Vlasov equation: $$v^\alpha\partial_{x^\alpha}f-v^\alpha v^\gamma\Gamma^\beta_{\alpha\gamma}\partial_{v^\beta}f=0$$ ...
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Diffeomorphism from $T M$ to $M \times \mathbb{R}^2$.

Let $M$ be a two-dimensional manifold. Suppose there are two vector fields on $M$, $X(p) = (p,V (p))$ and $Y (p) = (p,W (p))$, such that $$\text{span}\{V (p),W (p)\} = T_p M$$ for all $p \in M$. ...
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Tangent bundle is an oriented manifold

Recall that the tangent bundle $TM$ of a manifold $M$ consists of all pairs $(x, \overrightarrow{v})$ where $x \in M$ and $\overrightarrow{v}$ is the tangent space $T_xM$ of $M$ at $x$. Show that $TM$ ...
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Identity for projection and inclusion of the canonical one-form on the cotangent bundle

I'm currently reading through Introduction to Mechanics and Symmetry by Marsden and Ratiu, specifically the section on Cotangent bundles. I'm trying to do the following exercise: Let $N$ be a ...

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