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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).

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Othonormal Moving frame which is not adapted

Othonormal moving frame on ${\bf R}^3$ is a set $\{ e_i\}$ such that for any $p\in {\bf R}^3$, $$ e_i(p)\cdot e_j(p) = \delta_{ij}$$ When $M^2\rightarrow {\bf R}^3$, a frame $\{ e_i\}$ is adapted on $U=V\cap M$, $V\subset {\bf R}^3$, if for any…
HK Lee
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Partition of unity subordinate to an induced atlas on boundary of a manifold.

Let´s say we have a partition of unity $\{\chi\}_{i \in I}$ subordinate to $\{U_i\}_{i \in I}$ where $U_i$ can be found in the atlas $$\mathcal{A}_{M} := \{(U_i,\varphi_i):i \in I\}$$ on a smooth manifold $M$ with boundary $\partial M$. We get an…
Ben123
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If $f: M \to N$ is a smooth map between compact connected manifolds and $\operatorname{rank}{df} = \dim{N}$ then all pre-images are diffeomorphic

Let $M,N$ be compact connected manifolds, $f:M \to N$ a smooth map with $\operatorname{rank}{(df)}=\dim{N}$. Then for all points $p,q \in N$ ; $f^{-1}p$ is diffeomorphic to $f^{-1}q$. Please help me solve this question, I've no idea.
henry
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A question about contractions

In Lee's Introduction to Manifolds book in chapter 13 (page 335) they define a contraction $i_X: \Lambda^k(V) \to \Lambda^{k-1}(V)$ as $$i_X \omega (Y_1,...,Y_{k-1}) = \omega(X, Y_1,...,Y_{k-1})$$ First question I have is: Isn't there a typo and…
Math_Day
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scalar-flat metrics

Does anyone know if there exists a scalar-flat metric on the $n$-sphere, $n>4$, such that it is not Ricci flat. This should be easy, because it seems doubtful that spheres can carry a Ricci flat metric, but I'm having trouble proving existence.
Q-the-curvature
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Purpose of sectional curvature

I'm trying to get more knowledge on Riemann, Ricci and Einstein tensors and while reading Differential Geometry Curves Surfaces Manifolds Third Edition - W. Kühnel I came across the sectional curvature of the Riemann manifold with respect to the…
JackpotWizard 180
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A surface patch with compact image

Let us call surface patch a smooth function $x:U\to\mathbb R^3$ with domain an open set in $\mathbb R^2$ that is injective and has linearly independent partial derivatives at each point of its domain. The image of such a thing can be closed. For…
Mariano Suárez-Álvarez
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Proof of Existence of the Riemannian Density ( John Lee's Smooth manifodls)

I am reading the John Lee's Introduction to Smooth manifolds, Second Edition, Proof of Proposition 16.45 and stuck at understanding some statement : Proposition 16.45 (The Riemannian Densitiy ) Let $(M,g)$ be a Riemannian Manifold with or without…
Plantation
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Is this expression for the Riemann Tensor correct?

I've been given this exercise in my self studying differential geometry. Assume $f_1,\dotso,f_4$ be a local basis of vector fields on a manifold $M$, and let $\nu_1,\dotso,\nu_4$ be the dual basis. Assume that $[f_i,f_j]=p_{ij}\,^kf_k$ and that…
guido giuliani
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Doubt about the differential in differential geometry?

I have to answer the following question: Given $f:\Bbb{S}^2\to \Bbb{S}^2$ with $(x,y,z)\mapsto (\cos(30^{\circ} )x+\sin(30^{\circ})y,\sin(30^{\circ})x-\cos(30^{\circ})y,z)$ show that $df_p(v)\neq 0$ for all $v\in T_p \Bbb{S}^2$ I am a bit…
Red Banana
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Show two curves have constant lifting.

I want to prove that the curve $$\alpha(t)=(r\cos t,r\sin t,bt),~b>0,r>0$$ (which is a circular helix) and $$\beta(t)=(e^s,e^{-s},\sqrt{2}s)$$ both share the property of "constant lifting". I'm not sure how to mathematically say that, my first…
Fabrizio Gambelín
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$M$ $3$-manifold, $\omega$ is $1$-form defined on $M$, distribution $\ker \omega$ integrable, $\ker \omega = T \mathcal{F}$ foliation $\mathcal{F}$.

There is a several-part question I'm trying to do, which I'm hoping to get some feedback on the parts I've completed and some help on the parts I can't complete. Suppose $M$ is a compact connected $3$-manifold and $\omega$ is a nowhere zero $1$-form…
Math_Day
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Showing that $df_p\neq 0$ for all $p\in f^{-1}(0)$?

I'm trying to solve the following problem: Consider the function $f: S^2 \to \Bbb{R}$ given by $f(x,y,z)=x^{2023}+y^{2023}+z^{2023}$. Show that $df_p\neq 0$ for all $p\in f^{-1}(0)$. I'm thinking about this: Differentiate $f$ with respect to…
Red Banana
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Is there a conformal diffeomorphism between $\mathbb{R}^3$ and $S^1\times S^2$ minus a point?

Is there a conformal diffeomorphism between $\mathbb{R}^3$ and $S^1\times S^2$ minus a point on the $S^2$? I would have thought not, because the fundamental group of $\mathbb{R}^3$ is trivial, but that of $S^1\times S^2$ minus a point on the $S^2$…
user12588
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Integral curves of vector fields

Let us consider the vector field $X=x \frac{\partial}{\partial x}+y \frac{\partial}{\partial y}$. I'd like to compute the integral curves. Solving the system of ODE $\begin{cases} x'(t)=x(t) \\ y'(t)=y(t) \end{cases}$ we clearly get the curves…
TheWanderer
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