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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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Formula for composition of pull back

Problem says: $(GF)^{*}=F^{*}G^{*}$ , that is, for any form $\xi$ on $P$ , $(GF)^{*}(\xi)=F^{*}(G^{*}(\xi))$ So, by hint, I showed 0-form and 1-form cases. Let $f$ be a function on $P$ . Then $(GF)^{*}(f) = f(GF) = (fG)F = (G^{*}(f))F …
Darae-Uri
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Symmetricity of the extrinsic curvature tensor

I had referred to this structure earlier in a previous question which went unanswered. If $u$ and $v$ are in $T_pM$ and $n \in (T_pM)^\perp$ then the extrinsic curvature form $K$ be defined as, $K(u,v) = g(n,\nabla _u V)$ (where $V$ is a local…
Student
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Showing that a mapping is an isometry

Can anyone please help with this question? I have tried substituting but I'm not sure if that's correct so think I am missing something. Let S denote the surface of revolution $(x, y, z) = (\cos{\theta} \cosh {v}, \sin {\theta} \cosh {v}, v),$ $\ 0…
George
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basic question of differential geometry.

I'm taking a course on general relativity and I did a problem that I have now found out I have no idea how to interpret. The surface of a paraboloid has the metric $$ds^2=(1+r^2)dr^2+r^2d\theta^2$$ I was asked to do parallel transport of the vector…
DLV
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Is my proof of showing a helicoid and catenoid are isometric, correct?

This is the question I have: Let $S$ denote the surface of revolution $$(x,y,z)=(\cos\theta \cosh v, \sin \theta \cosh v, v)$$ $0 < \theta < 2 \pi$ and $-\infty < v <\infty$ and $S'$ the surface $$(x',y',z')=(u \cos \phi, u \sin \phi, \phi)$$ $0…
user232183
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Question on the definition of connection in Liviu.Nicolaescu's book

In "Lectures on the Geometry of Manifolds",Liviu said that a connection on a principle $G$-bundle defined by an open cover $(U_{\alpha})$ and gluing cocycle $g_{\alpha\beta}:U_{\alpha\beta} \to G$ is a collection $$A_{\alpha}\in…
C Weid
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Definition of "differential"

I am confused about the definition of "differential". Sometimes I see it is a pushforward mapping $df:TM\rightarrow TN$, which gives another tangent vector when acting on a tangent…
mmssm
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Example of a proper homotopy between smooth functions on manifolds

Let $h:S^{n-1}\to S^{n-1}$ be $C^{\infty}$ map. How to prove that a function $F: S^{n-1}\times[0,1]\to S^{n-1}$ given by $$F(v,t)=(\cos{\pi t})v+(\sin{\pi t})h(v)$$ is proper $C^{\infty}$ map?
dmm
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Integrals of forms are equal implies they differ by $d\mu$

The problem is about the proof of the following result. If $\omega_1,\omega_2 \in \Omega^n_c(X)$ (where $X$ is smooth manifold) are such that $\int_X\omega_1=\int_X \omega_2$ then there is $\mu\in\Omega^{n-1}_c(X)$ such that…
dmm
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What is "the line bundle $\Omega^n(M)$"?

In this Wikipedia article here what is "the line bundle $\Omega^n(M)$"? It seems to me that there can be many different line bundles on a smooth manifold $M$ so it's not clear to me what speaking of $the$ line bundle here means. Also, it's not…
self-learner
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Understanding the definition of a closed manifold

Let $D\subset R^n$ be a bounded domain with smooth boundary. Is $\partial D$ a closed manifold?
user331553
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Critical points of Exponential Map

How do you find critical points of an exponential map? I am working with a sphere of radius R. I know that an exponential map maps vectors of the tangent plane to a neighborhood of the point Q on the surface (in this case a sphere). I honestly have…
Aksel'sRose
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About cotangent bundle

$$\text{T}^*U\to\varphi(U)\times(\mathbb{R}^m)^*,\space(x,\lambda)\mapsto\left(\varphi(x),(D_{\varphi(x)}\varphi^{-1})^*(\lambda)\right)$$ What does $D_{\varphi(x)}\varphi^{-1}$ mean? Because I know "$D$" of sth, normally sth should be a function…
Upc
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Two curves with Same curvature but are not isometric.

question: Give an example of two space curves with the same curvature but are not isometric to each other(there is no isometry between them). I am using the Elementary differential geometry book by Presley. I really only know how to do curvature of…
MeowBlingBling
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Riemannian Distance

If $(M,g)$ is a Riemannian manifold, $g$ can be used to turn $M$ into a metric space. For two points $p,q\in M$, $d(p,q)$ is usually defined to be the infimum of the lengths of all possible piecewise smooth (or $C^1$) curves connecting $p$ and $q$.…
frog
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