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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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Writing an approximation of the parallel transport map in terms of the covariant derivative.

let $X,Y:M\to TM$ be vector fields on $M$. $\nabla_XY$ is the change in $Y$ along the flow curves of $X$. so for a point $p \in M$ let $\phi^X(t):\mathbb{R}\to M$ be a flow curve of $X$ passing through $p$ : $$(\phi^X)'=X \; \; ; \; \;…
Ziad H. Muhammad
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What is the torsion and curvature in modern language?

I think it can not be the torsion and curvature in the connection context, for these two are anti-symmetric in two variables, so that they must vanish on a one-dimensional space (tangent space of a curve). So what should these two notions be in a…
Anonymous Coward
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Geodesics on oblique helicoid using isometry

I am trying to find the non trivial geodesics on an oblique helicoid(non-minimal) surface apart from the rulings. Instead of using the usual geodesic equations and also because I was interested in seeing the correspondence between geodesics through…
Vishesh
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Properties of Unit Tangent Bundle

If $(M, g)$ is a Riemannian manifold with or without boundary, its unit tangent bundle $UTM := \{(p, v) \in TM \mid |v|_g = 1\}$ is a smooth, properly embedded codimension-1 submanifold with boundary in $TM$, with $\partial(UTM) = \pi^{-1}(\partial…
James C
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First fundamental form and angle between curves

On surface, for which $$ds^2=du^2+dv^2$$ find the angle between lines $v=u$ and $v=-u$.This exercise is related to the first fundamental form. I think I need to find the angle between curves with parametrization: $$u(t)=t, v(t)=t$$ and $$u(s)=s,…
Andrew
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Proof that the vector area is the same for all surfaces sharing the same boundary

A proof of the title was posted on physics stack exchange asking for explanation of the steps, I had written an explanation of the steps as an answer. However, after writing this answer there has been something which has been bothering me about the…
tryst with freedom
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Symmetry of a curvature tensor

Could you help me with the following problem, please? I have the next proposition of the book An Introduction to Riemannian Geometry with Applications to Mechanics and Relativity of Leonor Godinho and José Natário. Proposition 1.8 If $M$ is…
Cal22
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Does unit ball in homogeneous Riemannian manifold $M$ translates isometrically to another unit ball in $M$?

Definition of Homogeneous Riemannian manifold: Pick a point $p \in M$. For all $q \in M$, there exists $\phi \in ISO(M)$ such that $\phi(p) = q$. Take Riemannian metric $d$. Then, for arbitrarily small $\delta > 0$, can I assert that $B(p, \delta)$…
James C
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Existence of local vector field extension of a vector with vanishing covariant derivative in all directions at that point

While there generally exist on differentiable manifolds $ M $ and for a choice of a torsion-free connection $ \nabla $ locally geodesic vector fields $ V \in \mathfrak{X}(M) $ extending a given (non zero) vector $ v \in TM $ in the sense that $…
Septillion
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Is there a closed curve with constant radius of curvature？

Is there a closed curve with constant radius of curvature？ Circle is an answer. Is there some other solution in 3D?
arax
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Bracket $[X,Y]$, where $X$ and $Y$ are differentiable vector fields.

Let $X$ and $Y$ be linearly independent vector fields in $\mathbb{R}^3$ and let $\omega$ be a 1-form whose kernel (for each point q of $\mathbb{R}^3$ ) is the plane generated by $X (q)$ and $Y (q)$. Show that the condition $d\omega\wedge\omega = 0 $…
Karly
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Integrating a smooth family of differential forms

I have been trying to find a reference for the following : Suppose I have a smooth family of differential forms $\omega_t$ that are exact. Then we can find a smooth family of differential forms $\mu_t$ such that $d\mu_t=\omega_t$. I have been told…
Someone
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Bonnet's theorem in do Carmo

I'm working on problem 2 in section 5-4 of do Carmo's "Curves and Surfaces" (2nd edition), which requires us to show that if S = (x,y,f(x,y)) is a complete non-compact regular surface, then the minimum curvature if we go sufficiently far out, will…
Sina
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how to calculate the curvature

The Question is: Let $\gamma(t)$ be a regular curve lies on a sphere $S^2$ with center $(0, 0, 0)$ and radius $r$. Show that the curvature of $\gamma$ is non-zero, i.e., $κ \ne 0$. My question: We define $k$ as $k=\frac{(||\gamma'' \times…
louis
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Find local isometry $f:X\rightarrow Y$,

So I'm revising for an exam in differential geometry, and have got stuck on a question regarding isometries. We define a local isometry as a map $f:X\rightarrow Y$, (between 2 regular m-surfaces X,Y) where, for all $p\in X$ and all $u \in T_{p}X,…
Olivia77989
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