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Questions tagged [differential-geometry]

Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses on "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such structures, use (differential-topology) instead. Use (symplectic-geometry), (riemannian-geometry), (complex-geometry), or (lie-groups) when more appropriate.

Differential geometry is a branch of mathematics that uses techniques in calculus and linear algebra to study geometry. It is closely related to differential topology and PDEs (geometric analysis).

32835 questions
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Natural connection on tautological bundle over real Grassmannian

Let me get to the point immediately: Is there a natural connection on the tautological vector bundle over a Grassmannian (of a real vector space equipped with an inner product)? In a paper I'm reading there is a smooth 1-parameter family of…
Daan Michiels
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Why is the parallel transport on the 2-sphere SO(3)-equivariant?

I am trying to prove the following equation for $R \in SO(3)$: \begin{equation*} R^{-1}P_{R(\gamma)}(R(v)) = P_{\gamma}(v) \end{equation*} where $\gamma \colon \lbrack 0,1 \rbrack \longrightarrow S^2$ is a curve and $P_{\gamma}$ the parallel…
TerranMathematician
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The extension of Manifold

If $M \subset \mathbb R^n$ is a compact smooth manifold with boundary, and ${M_\varepsilon }$ is the closed $\varepsilon$-neighborhood of $M$ in $\mathbb R^n$, then whether for sufficiently small $\varepsilon$, ${M_\varepsilon }$ is a smooth…
Summer
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diffeomorphism of the bundle chart of a Tangent bundle

I am studying about tangent bundle from the book "J. M lee", on page 66 of the book a map $\tilde{\phi} : \pi^{-1}(U) \to \mathbb{R}^{2n}$ is defined by $$v^i \frac{\partial}{\partial x^i}|_p \mapsto (x^1(p),\cdots, x^n(p),v^1,\cdots,v^n)$$ where…
user103005
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Example of a sum of complete vector fields

Can anyone give an example of two vector fields $X_1$ and $X_2$ which are complete but their sum $X_1+X_2$ is not complete?
user146507
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Are any two Cantor sets ; "Fat" and "Standard" Diffeomorphic to each Other?

All: I know any two Cantor sets; "fat" , and "Standard"(middle-third) are homeomorphic to each other. Still, are they diffeomorphic to each other? I think yes, since they are both $0$-dimensional manifolds (###), and any two $0$-dimensional…
gary
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connection on tensor bundle

I know that if $E$ and $F$ are two vector bundle with connection $\nabla^E$ and $\nabla^F$, then it is natural to define tensor connection $\nabla = \nabla^E\otimes 1 + 1 \otimes \nabla^F$ on $E \otimes F$. By I have a stupid question: is is…
Lelouch
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Reconstruct a curve from its curvature and torsion

Given two smooth functions $k(s)$ and $\tau(s)$, how can we construct a curve in $R^3$ that its curvature is $k$ and torsion is $\tau$? Is there any conditions that $k$ and $\tau$ must satisfy as the curvature and torsion of a curve? It seems to be…
user47719
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Conditions that torsion is zero in a space curve

What are the conditions for torsion to be zero other than having a plane curve? The only thing I can thing of is an equation that have the torsion that cancels out each other.
vener
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Checking a map is an immersion

Let $f: \mathbb{R} \rightarrow \mathbb{S}^{1} \times \mathbb{S}^{1}$ given by: $f(t)=(\exp(it),\exp(qit))$. I want to show that $f$ is an immersion. OK I know the definition: we compute its derivative and check it is injective. My question is: can…
user17182
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Intersection points in a one-parameter family of lines

Given is a one-parameter family of lines, $$L(t) = \{ a(t) + \lambda w(t) : \quad \lambda \in \mathbb{R} \}$$ in which the base point $a$ and the direction vector $w$ vary smoothly with a parameter $t$ (you may assume that $|w| = 1$ and $w' \neq…
koletenbert
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The set of smooth maps from exotic smooth manifolds to the reals

Here a $M,N$ are topological manifolds and $\mathcal{A}$ and $\mathcal{B}$ are atlases. The brackets $[]$ denote the formation of the equivalence class of atlases. Let $(M,[\mathcal{A}])$ and $(N,[\mathcal{B}])$ smooth manifolds exotic to each other…
w_w
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property of a curve $\alpha(t)$

Find a parametrized curve $\alpha(t)$ whose trace is the circle $x^2 + y^2 = 1$ such that $\alpha(t)$ runs clockwise around the circle with $\alpha(0) = (0, 1)$. A parametrized curve $\alpha(t)$ has the property that its second derivative…
jigja
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Orientability of the sphere

how does one explain the following: "The sphere can be covered by 2 open sets using stereographic projection in such a way that the intersection of these 2 sets is a connected set $W$.Let $p \in W$, if the jacobian of transitions maps is negative…
Jr.
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Formula for torsion convention

Our class is using Do Carmo's Differential Geometry and I noticed a convention that Do Carmo seems to break when dealing with torsion. The background is 1.5.1 (what I was working on): Given $\alpha(s) = (a$ cos$(\frac{s}{c}), a $ sin$…
Lost
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