Questions tagged [tangent-bundle]

Question 1

For vector field $X, Y$ and one-form $\omega$ with its value in $T_M$, why does the identity $[\omega, \omega](X, Y) = \frac{1}{2}\{[\omega(X), \omega(Y)] - [\omega(Y), \omega(X)]\}$ hold?

Question 2

In Section 10.2 of Loring W. Tu’s Differential Geometry: Connections, Curvature, and Characteristic Classes, he explains how to patch connections on a vector bundle using partitions of unity. He uses ...

Question 3

I have this exercise: Consider the map $$ F: \mathbb{R}^{n+1} \times \mathbb{R}^{n+1} \to \mathbb{R}^{2},\ \ F(p,q) = (||p||^{2}, \left<p,q\right>). $$ Prove that $F$ is differentiable and $(1,...

Question 4

Let $T S^2$ denote the tangent bundle of the 2-sphere $S^2 = \{x \in \mathbb{R}^3 : \|x\|=1\}$. In this paper, Fodor proves that $T S^2$ is parallelizable, using machinery that I do not understand ---...

Question 5

Let $X$ be the geodesic vector field in $T^{1}\mathbb{R}^{2}$ and let $Y=(0,1)$ be the constant vector field on the fiber component of $T^{1}\mathbb{R}^{2}$. Then, integral lines of the vector field $...

Question 6

Like the title suggest, I'm having trouble in Michael Spivak first book about differential geometry, more specificaly chapter 3. An important note is that in his book, Spivak say that a manifold is a ...

Question 7

For each $x\in S^3$, where $S^3$ is identified with the unit quaternions, let $f^x_{h,l}:\mathbb{H}\rightarrow \mathbb{H}$ be the transformation $f^x_{h,l}(v)=x^hvx^h$. Taking the open sets $U_1=S^4\...

Question 8

In a certain exercise I have been asked to show that, given a smooth manifold $M$, the vector field $X: M \rightarrow TM$ such that $\forall p \in M$ the field gives the zero vector in $T_pM$ is an ...

Question 9

Let $M$ be a 2-dimensional smooth, orientable manifold in which there is a non-vanishing global vector field, that is an $X\in\mathfrak{X}(M)$ such that $X(p)\neq0,\,\forall p\in M$. Show that $M$ is ...

Question 10

Let $p:M\to X$ be a smooth vector bundle of rank $n$, and let $\tau_M$ be the tangent bundle of $M$. Since the bundle is smooth, we can take the derivative at any point of $M$, $d_mp: T_mM\to T_{p(m)}...

Question 11

Let $X$ be a smooth manifold with an almost complex structure $J$. Then does $J$ induces an almost complex structure (thus, a map $TTX\to TTX$) on the tangent bundle $TX$, canonically so that the zero ...

Question 12

I'm working through the proof of the Tubular Neighborhood Theorem and came across this decomposition of the tangent space of the normal bundle: $$T_{(x,0)}(NM) = T_{(x,0)}M_0 \oplus T_{(x,0)}(N_xM).$$ ...

Question 13

Setup. Consider the $n$-dimensional complex projective space $\mathbb{P}^n$, and consider the Euler sequence $$ 0\to \mathcal{O}_{\mathbb{P}^n}(-1) \to \mathcal{O}_{\mathbb{P}^n}^{\oplus n+1} \to T_{\...

Question 14

So I've confused myself on calculating the tangent space of $\mathbb{P}^n$. The definition of a tangent space I am working with is the set of all derivatives at $0$ of a smooth curve at a point $p$. ...

Question 15

I'm working through differential geometry specifically as it applies to general relativity. Many of the texts or presenters make the assertion that we always use an implicit or non embedded ...

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Questions tagged [tangent-bundle]

On vector field. [closed]

Patching connections with partitions of unity preserves torsion?

The tangent bundle of $\mathbb{S}^{n}$ is a connected manifold

Explicit proof that tangent bundle of the 2-sphere, TS^2, is parallelizable

Vector field tangent to hyperbolic circle

Construction of the topology on the tangent space of a smooth manifold $M$ in Michael Spivak way with curves

Splitting of the tangent bundle of a disk bundle

Tangent bundle and inmersion

Orientable surface with non-vanishing vector field Is Parallelizable

Vertical bundle is the square of the original bundle

Is the tangent bundle of an almost complex manifold also almost complex?

Why $T_{(x,0)}(NM) = T_{(x,0)}M_0 \oplus T_{(x,0)}(N_xM)$? (Theorem 6.24, Introduction to Smooth Manifolds, John Lee)

Zero locus of a global section of $T_{\mathbb{P}^n}(-1)$

Tangent space of the real projective space.

Implicit (non embedded) tangent spaces and tangent bundles in physics

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