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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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Nontrivial second order parallel symmetric tensor field on a general Riemannian manifold？

Some literatures show that for some special manifolds (e.g. space forms, Sasakian mfd. etc.), the second-order parallel symmetric tensor fields are constant multiple of the associated metric tensor. Are there nontrivial second order parallel…
Geom Zari
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Laplace’s Equation in Hyperspherical Geometry

I've been reading this reference. I agree with everything they say but there's something that I can't really understand...They get that, for example n=2, the potential created by some source on a sphere is zero at $\theta=\pi/2$ and when…
PML
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exact form on the unit circl

How can I show that $\alpha$ is exact form on $S^1$ iff $\int_{S^1} \alpha =0$? I used Stokes theorem to prove one direction. Could you please help me in the other direction?
saba
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Existence of minimizing geodesics

Let $ M $ be a Riemannian manifold with metric $ ds^2 $. Suppose that every two points in $ (M,ds^2) $ can be joined by a minimizing geodesic. Now let $ \mu ds^2 $ be a metric on $ M $ conformal equivalent to $ ds^2 $. Is it true that every two…
user55449
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Realizing any vector in tangent space as tangent vector

Let $S$ be a surface in $\mathbb{R}^3$. Let $U\subseteq \mathbb{R}^2$ be open (connected) and $\sigma:U\rightarrow S$ be a homeomorphism into $\sigma(U)\subset S$, with $\sigma(U)$ open in $S$, and $\sigma$ is smooth map. Assume that…
Beginner
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Computing the principal curvatures and directions without making reference to a parametrization

Consider, for instance, the catenoid given by $x^2+y^2 = \cosh^2 z$, and suppose that we want to find the principal curvatures and directions at some point $p=(x,y,z)$ of the surface. Of course, we can do this by parametrizing the catenoid as…
MSDG
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Does a manifold with two disjoint compact boundaries have two disjoint collared neighbourhoods?

Take a Hausdorff manifold $M$, with a disjoint boundary $\partial M$ composed of $\partial M_1$ and $\partial M_2$, such that the boundary is compact (I think this doesn't hold for non-compact one, with $\mathbb{R}^2$ with $|y| > x^{-1}$ removed as…
Slereah
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given first fundamental form, what type of surface?

In Pressley book I found problem: Find the first fundamental form and determine what kind of surface is patch $\sigma=\langle u-v,u+v,u^2+v^2\rangle$. It is easy to see that the first fundamental form is $(2+4u^2)du^2+8uv\,du\,dv+(2+4v^2)dv^2$. The…
pabodu
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How to show that a collection of charts is an atlas?

If I look at the n-sphere $$M = S^n = \{x \in \mathbb{R}^{n+1} : x_1^2 + x_2^2 + \ldots + x_{n+1}^2 = 1\}$$ with the subspace topology of $\mathbb{R}^{n+1}$. Then a chart for $1 \leq i \leq n + 1$ and $\epsilon = \pm 1$ is defined: $$U^{\epsilon}_i…
user582360
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How much area can I see on average if I’m hovering over a deformed sphere?

I’m interested in getting some estimates for how much area I can see in average if I’m above (let’s say height $h$) of a surface diffeomorphic to a sphere. Let’s start with a circle of radius $r$. If I’m on a height $h$ (i.e. on a distance of $h+r$…
Andreas Rüdinger
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Lie brackets in Eculidean spaces

Let $S^n\subset \mathbb{R}^{n+1}$ be the $n$-sphere, and $X, Y$ be vector fields on $S^n$. My question is, how to compute the Lie bracket $[X, Y]$ without using local coordinates? I've seen in a note that $X, Y$ can be viewed as maps $X, Y:S^n\to…
string
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Nieh-Yan term in the mathematics literature?

Let me establish my notation by stating the Cartan structure equations for the frame bundle of a Riemann 4-manifold. The curvature two-form is defined in terms of a ${\rm SO}(4)$ frame bundle connection two-form ${\omega^a}_b$ $$ {\bf R^a}_b =…
mike stone
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Fibre Hopf Fibration diffeomorphic to $S^1$

I have been given the map $ h: S^3\to S^2 $ Given by $h(z_0,z_1)=(|z_0|^2-|z_1|^2,2i\overline{z_0}z_1)$. I have proved that this map is a well-defined smooth submersion. Next is to show that the fibres of $h$ are an embedded submanifold of $S^3$…
user408856
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Is there any difference between $\mathbb{R}^3$ and Euclidean space denoted $\mathbb{E}^3$?

I just can't seem to make any distinction between the two. If anyone has a simple explanation I'd like to hear it.
MRT
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Why Lie derivative provides a representations of a Lie algebra

I am reading a book about general relativity. After introducing the Lie derivative and Lie bracket, the author claims that the Lie derivative provides a representation of the Lie algebra of vector fields from the relation $$[L_V,L_V]=L_{[V,W]}$$ I…
Craig Thone
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