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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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Why is it the metric that allows for the canonical tangent space/ cotangent space identification?

I have seen the phrase "The metric allows for a canonical identification of the tangent space with the cotangent space" all over diff geo resources and questions. I understand the map and why it serves as an identification, but since it works using…
Marc Mekhanik
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Need help with the proof of Ricci tensor of Einstein and anti self dual manifold.

Let M be an anti-self dual and Einstein 4-manifold with scalar curvature s. Then the Ricci tensor $c_Z$ of the twistor space is given by $$c_Z (E,E) = (s/4 - t(s/12)^2) \|X\|^2 + (1+ (ts/12)^2)\|V\|^2,$$ where $X= \pi_\ast E, V= \mathcal{V}E$ and…
SkyWalker
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cycles in gradient descent on non-euclidean manifold?

In a euclidean space, we cannot have gradient descent of a function $f$ and still have cycles. i.e. if $x(t)$ is the path traced by the dynamical system given by $\dot x(t)=-\nabla_x f(x)$, for some function $f$, then we cannot have…
user56834
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Geodesic on a surface of revolution using Christoffel Symbols.

I have the following problem: For a function $f:[a,b]\rightarrow \mathbb{R}_{>0}$ and for the open set $U=\{(u_1,u_2)\vert\: a
Tom
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Is it possible that a 2-sphere has a Weitzenböck connection?

I mean, is it possible to have a connection on the 2-sphere with vanishing curvature but not vanishing torsion? In a more general sense, it is know that every Riemannian manifold has a Levi-Civita connection,is this true for the Weitzenböck…
yess
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$k$-forms as modules over $C^\infty(M)$

Let $M$ be an $n$-dimension manifold. Is $E^k(M)$ a finite dimension module over $C^\infty(M)$? Here $E^k(M)$ is the space of $k$-form on $M$, $k
Qijun Tan
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Tolman-Bondi-Lemaitre space times

One can see this reference for TBL space-times. I would like to know how the explicit expression for the function called $G$ in equations $3.108,3.108,3.110$ in the above reference is obtained. Also it would be nice to see some further references…
Student
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Vector field decomposition into gradient and hamiltonian vector field

I have just read (without further explanation) that any vector field $(v_x(x,y),v_y(x,y))$ from $\mathbb{R}^2$ to $\mathbb{R}^2$, which has a continuous derivative, can be uniquely written as the sum of a gradient and a hamiltonian vector field,…
Dirk Dinther
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Torsions of Asymptotic Curves

Let $p$ be a hyperbolic point of a surface $S$. Let $\alpha_1$ and $\alpha_2$ be two asymptotic curves passing through $p$ (in two different asymptotic directions) and assume that they have nonzero curvatures at $p$. Prove that if $\tau_1$ is the…
presto
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why do we need $4$ dimensions to embed a two dimensional shape on a surface

My Lecturer mentioned at the beginning of my differential geometry course that you need at least $4$-D to embed a $2$-D shape on an ambient space ( not too sure what ambient space means ....) My question is why is this so ? We're only a few weeks…
excalibirr
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Gaussian curvature of the pseudosphere

I am asked to show that the pseudosphere has Gaussian Curvature $-1$ at all points. So I parametrized the tractrix as: \begin{equation} \alpha(t) = (\sin{t},\cos{t}+\log{\tan{\frac{t}{2}}}) \end{equation} So the surface of revolution would…
davidaap
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Restriction of the Laplace Beltrami operator

Given the expression of the Laplace-Beltrami operator $\Delta M$ on a Riemannian manifold $M$ , is there any method for determining the expression of the Laplace-Beltrami operator $\Delta N$ where $N$ is a submanifold of $M$ . Actually I am…
Riadh
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Prove curvature of sphere curve $\geq R^{-1}$

Let $\bf x(s)$ be a sphere curve lying on the surface of a sphere with center $\bf{p}$ and radius $R$ satisfying $$|\bf{x}(s)-p|^2=R^2.$$ I want to prove that the curvature $\kappa \geq R^{-1}$. I assumed the center is the origin as this does not…
user30523
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Is $|x|$ a smooth manifold?

The function $y = |x|$ is not a differentiable function from $\mathbb{R} \rightarrow \mathbb{R}$. But considering the graph of $y = |x|$ as a subspace of $\mathbb{R}^2$, we can endow this space with a smooth structure consisting of the single smooth…
Frederic Chopin
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Curvature function and rate of change of angle

Let $\gamma:(a,b)\rightarrow \mathbb{R}^2$ be a smooth curve with $\| \dot{\gamma}(s)\|=1$ for all $s\in (a,b)$. Fix $s_0\in (a,b)$ and let the unit vector $\dot{\gamma}(s_0)$ be represented by $(\cos \phi_0,\sin\phi_0)$. Then there is smooth…
Beginner
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