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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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Showing that every smooth 1-form is an immersion

I am reading the John Lee's Introduction to smooth manifolds, Proposition 22.12 and stuck at showing that every smooth 1-form $\sigma : M \to T^{*}M$ is an immersion : I am trying to understand the underlined statement. Why can we deduce that…
Plantation
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Covariant derivative commutes with trace (contraction)

I'm trying to solve Exercise 4.3 in J. Lee's Riemannian Manifolds book. The setting is as follows: let $\nabla$ be a linear connection on $M$. I want to show property (b) of Lemma 4.6, which says that $\nabla$ commutes with all contractions. To make…
Sha Vuklia
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In the proof of the Proposition 18.16 of John Lee's smooth manifolds ( Equivalent condition for the invariance of $\tau$ under the flow of $X$ )

I'm reading John M. Lee's Introduction to smooth manifolds, First Edition, Proposition 18.16 and some question arises : Why the underlined statements are true? ; i.e., why the map $T$ is smooth? Under what smooth structure on $T^{k}(T_pM)$ ? And…
Plantation
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Zero of the lie derivative of the hopf fibration

I lack the intuition to understand the answer of the following problem. Let the so called hopf fibration given by $h(x,y,z,t)=(x^2+y^2-z^2-t^2),2(yz-xt),2(xz+yt)$ and consider a vector field $X$ on $S^3$, namely…
Bessel
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The natural injection $\mathbb S^k\longrightarrow\mathbb S^n$ is an embedding

Show that $f:(x_1,...,x_k)\in\mathbb S^k\mapsto(x_1,...,x_k,0,...,0)\in\mathbb S^n$ is an embedding. I am new to differential geometry, and I have a hard time wrapping my head around this one. The answer should be really trivial but I can't quite…
maxjw91
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Diffeomorphism of $\Bbb R^{n}$ to $\Bbb R^{n}$ which sends lines to lines is affine

Let $f:\mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ ($n \geq 2$) be a diffeomorphism which preserves lines (i.e. the image of any line is a line). Prove that $f$ is an affine map. The idea is to prove that the Jacobian matrix is constant and I can…
Steppewolf
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Extension of Shape operator into a Riemannian manifold

When majority of authors textbook of differential geometry (like a O'Neill, Boothby, and some PDFs from Google) deal with a shape operator of $M$, I have never seen that the shape operator is defined on a manifold and solely have seen the surface…
Hanwoong Cho
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Clarification on proving that $A(x,y,z)=(-x,-y,-z)$ from $S^2\to S^2$ is a diffeomorphism?

I am trying to prove the following: So I think I must prove first that $A$ is differentiable. I am looking at the following definition: I don't understand where the $\Bbb{R}^2$'s (or $\Bbb{R}^3$) appear here. I thought about the following…
Red Banana
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Partial derivative of one component of coordinate system in terms of another

I'm reading through Spivak's Comprehensive Introduction to Differential Geometry, vol. 1. On page 35 I encountered the following: Let $(x, U)$ be some coordinate system with $x(p) = (x^1(p), \ldots, x^n(p))$ and $f:M \to \mathbb{R}$ be a function.…
Tölvukisa
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Proof that $S^2$ is not parallelizable

I'm trying work out a proof from Arnold's Ordinary Differential Equations that the sphere $S^2$ is not parallelizable. First, he sets as an exercise (page 298) to determine which of the manifolds below are equivalent by…
user804930
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Motion of curves vs Motion of vector fields?

I am trying to understand the geometry of curves and surfaces, in particular, surfaces that are obtained through the motion curves. Let us consider the space curve $\gamma :I\rightarrow $ $\mathbb{R}^{3}$ such that $\gamma =\gamma \left( s,t\right)…
ruudvaan
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Pullback metric on embedded manifolds

I study the pullback of a map $f$, in special the pullback metric from manifold $\mathcal{Y}$ on a base smooth manifold $\mathcal{X}$. I understand that: the push forward operator $f_*: T_x \mathcal{X} \to T_y \mathcal{Y}$ maps every vector $a \in…
Bruno Peixoto
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Show that if $f(x,y)\geq a(x^2+y^2)$ for some $a > 0$, $f$ is smooth and $f(0,0) = 0$, then Gaussian curvature at $(0,0)$ positive

There is a question in Differential Geometry which I am not sure how to solve: For $f:\mathbb{R}^{2}\to\mathbb{R}$, show that if $f(x,y) \geq a(x^2+y^2)$ for some constant $a > 0$, $f$ is a smooth function, and $f(0,0) = 0$, then Gaussian curvature…
ExtraFlash
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Poisson bracket useful formula

Equation 1.2.10 of Wolf's "Spaces of constant curvature" is: $[fX,gY] = fg[X,Y]+f(Xg)Y -g(Yf)X$ where $X$ and $Y$ are vector fields, and $[-,-]$ is the Poisson bracket. I assume $f$ and $g$ are real valued functions, so $Xg$ and $Yf$ are also real…
Oldlag
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Integral of a compactly supported exact n-form (on a not necessarily compact n-manifold)

Let $M$ be an oriented n dimensional manifold and let $\eta$ be an exact n-form on $M$ (say, $\eta=d\omega$). If $M$ is compact, I know that Stokes theorem implies that $\int_M \eta=0$ (if I understand correctly, this is by "promoting" $M$ to a…
Lostphysicist
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