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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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Does existence of two Killing vectors $X,Y$ with $[X,Y]\neq 0$ imply existence of a third linearly independent Killing vector?

Suppose we have a Riemannian or Lorentzian manifold with two Killing vector fields $X,Y$ such that $[X,Y]\neq 0$. Does this imply existence of a third linearly independent Killing vector field $Z$? I know that $[X,Y]$ is itself a Killing vector.…
Kuba
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products of manifolds with boundary

I am struggling a little bit with the following problem. Let $M$ and $N$ be smooth manifolds of dimension $m$ and $n$, respectively. Show that the product $M\times N$ is a smooth manifold of dimension $m+n$. Is this also true for smooth manifolds…
user223794
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Need help understanding a lift of a vector field

This is a question from my differential geometry assignment: Let $\pi:M\to N$ be a submersion between two smooth manifolds and $X\in \Gamma(TN)$ is a vector field. We need to show that there is a smooth vector field on $M$ that is $\pi$-related to…
ZP R
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How do I apply smooth $p$-forms to vector fields?

Let $M$ be a smooth manifold. My definition of a smooth $p$-form is a map section $\omega: M \rightarrow \Lambda^p TM^*$, i.e. if $q \in M$ is contained in a chart $U$ with co-ordinates $x_1, \ldots, x_n$, then $$ \omega (q) = \sum_{i_1 < \ldots <…
Paul Slevin
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A proof for the Mobius Strip parametrization

According to Elementary Differential Geometry by A N Pressley, a parameterization for Mobius strip is : $\textit{Example 4.9}$ The Möbius band is the surface obtained by rotating a straight line segment $\cal L$ around its midpoint $P$ at the same…
user200918
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What is the shortest path equation between 2 points on a cone?

What is the shortest path equation between 2 points (from A to B) on a cone surface? $A= (x_1,y_1,z_1)$ and $B=(x_2,y_2,z_2)$ and cone equation is $x^2+y^2=r^2z^2$ I know that the shortest path is a line on the cone surface as shown below but I…
Mathlover
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vector field on $\mathbb{R}^n$ versus on manifold

I am looking for a counter example that why the $\mathbb{R}^n$ definition of vector field fail on a manifold. The following is a summary of what I learnt few years ago. Start with the idea of differentiating vector fields in Euclidean space: choose…
math101
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Intuition about geodesic incompleteness

To state the context, I am familiar with the Hopf-Rinow theorem. My request is three fold, I would like to know of general classes of geodesically incomplete spaces. I basically want to see lots of examples for this. I want to know of techniques…
Student
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equivalent definitions of tensorfields

I have been confused about something for quite some time now and I would really much appreciate to get a clear explanation of the following. There are two equivalent definitions of how to define tensor fields on a manifold $M$. The first is as…
harajm
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principal curvature of the flat torus

I am looking at the Hopf-fibration and I am looking at the preimage of the equator in $\mathbb{S}^2$. I think that I have proved that this is just the flat torus and now I want to calculate the principal curvatures of this torus. My general approach…
harajm
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Concrete example of vector field along a map

In this book the definition of a vector field along a map $f: M \to N$ is given as follows: I am currently trying to understand this definition. For this purpose I wanted to work out a concrete example. But I need some help. Here is the…
a student
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Riemannian geometry vs Hyperbolic geometry

I am learning differential geometry in this semester. Concerning the riemannian geometry, if the cross-sectional curvature (riemannian metric ) is negative at every point, the manifold which arises is hyperbolic. At the other hand hyperbolic…
ivo
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The definition of vertical bundle (coordinate-free reasoning)

Let $\tau_Q \colon TQ \rightarrow Q$ be tangent bundle, $T\tau_Q : TTQ \rightarrow TQ$ be its derivative. $VQ$, defined as $\mathrm{\mathop{Ker}}\; T\tau_Q$, is a subbundle of the second tangent bundle $\tau_{TQ}\colon TTQ \rightarrow TQ$, and $VQ$…
akater
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$(e^x \cos y,e^x \sin y)$ is local diffeomorphism not global

Let $f:\mathbb{R^2} \to \mathbb{R^2}$ defined by $f(x,y)=(e^x \cos y,e^x \sin y)$ I have showed that $f$ is a local diffeomorphism by using inverse function theorem, that is $\det(Df)=e^x \gt 0$ for all $x$, so $Df$ is invertible, hence local…
SamC
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When are the eigenvalues of the second fundamental form equal to the principal curvatures?

I am confused about the following concerning the second fundamental form. Consider a surface $S$ $\subset R^3$ If we consider a chart at a point $p \in S$, $f$: $R^2$$\to S$ and suppose $\partial f/\partial x$ and $\partial f/\partial y$ are…
TheGeometer
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