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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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Taylor expansion of Christoffel symbols

I would like to show that in geodesic coordinates with origin $p$ $$\Gamma_i = -\frac 12 \sum_j \mathrm{R}_p(\partial_i, \partial_j) x^j,$$ where $\Gamma_i = \Gamma_{ij}^k \partial_k \otimes \mathrm{d}x^j$ and $\mathrm{R}_p$ is the Riemann tensor at…
Kofi
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Unit-speed reparametrizations

I have to prove that for a regular parametrized curve there is essentially (up to sign and a constant) a unique reparametrization which makes it a unit-speed curve. Let $x$ be a curve, $s(t) = \int_{t_0}^{t} \left \| \frac{\mathrm{d} x}{\mathrm{d}…
user14174
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Revolution surfaces of constant Gaussian curvature

I'd like help with the following question: Prove that all revolution surfaces $(\phi(v) \cos u ,\phi(v) \sin u,\psi(v)) $ of constant Gaussian curvature $k = -1$ is one of the following types: $\phi(v)=C\cosh v$ and $\psi(v)=\int_0^v…
Jr.
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What is wrong with this exercise in do Carmo's Differential Geometry?

This is an exercise in do Carmo's Differential Geometry: Let $\alpha : I \longrightarrow S$ be a curve parametrized by arc length $s$, with nonzero curvature. Consider the parametrized surface \begin{align}\textbf{x}(s,v)=\alpha(s)+vb(s), & s \in…
user9464
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Gaussian curvature and mean curvature sufficient to characterize a surface?

Is the knowledge of Gaussian and mean curvature (and thus of the principal curvatures) sufficient to characterize a surface uniquely? If not, is there another geometric quantity one can add to obtain a unique characterization?
madison54
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Techniques to check if a surface is regular

This is (a part) of exercise $7c$ page $66$ of DoCarmo's book (Differential geometry of curves and surfaces). Let $f(x,y,z)=xyz^{2}$. I'm trying to figure out if the preimage of $f$ under $0$ is a regular surface. Basically the preimage is the union…
user17182
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(Determinant of) Hessian in local coordinates

Let $f\colon M\to \mathbb{R}$ be a smooth function on a manifold $M$ with a critical point $p$. We define its Hessian at $p$ via $H(u, v)=(UVf)(p)$ where $u, v\in T_pM$ and $U$ and $V$ are vector fields with $U_p=u, V_p=v$. I wonder if there is any…
EPS
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Volume form on S^1

I know that the volume form on $S^1$ is $\omega= ydx-xdy$. But how I can derive that? The only things that I know are the definition of differential q-form, and the fact that the vector field $v= y \frac{\partial}{\partial…
andreasvr
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Divergence in spherical coordinates problem

I have this formula for the divergence of a vector field: $$\nabla_m V^m = \frac{1}{\sqrt{|g|}} \frac{\partial (V^m\sqrt{|g|})}{\partial x^m}$$ The metric tensor in spherical coordinates: $$ g=\begin{pmatrix} 1 & 0 & 0\\ 0 & r^2\sin^2(\theta) & 0\\…
BinaryBurst
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applications of viewing some manifolds as homogeneous spaces

I'm doing some reading on isometric actions and the book I'm following just proved that one has diffeomorphisms $\mathbb{S}^{n} \cong \mathrm{SO}(n+1)/ \mathrm{SO}(n)$, $\mathbb{R}P^{n}\cong \mathrm{SO}(n+1)/S(O(n)\times O(1))$. Then he proves…
Sak
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Why is connection a map from $\Gamma(E)$ to $\Omega^1\otimes\Gamma(E)$?

On the site Vector Bundle Connection, it gives two definitions of a connection. One is view a connection as a linear map from a section of $E\otimes TM$ to a section of $E$: $$ D:\Gamma(E\otimes TM)\rightarrow\Gamma(E) $$ I can understand this…
hxhxhx88
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Mixed Partials from Peter Petersen's book

I am trying to understand how mixed partials are defined for a function $\gamma : \mathbb R^m \rightarrow M$, where $M$ is an $n$ dimensional manifold, from Peter Petersen's "Riemannian Geometry" (Page 112). Please refer to the book. Let…
April
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Why the connected sum is a differentiable manifold

maybe this is a stupid question. I know how to prove that the connected sum $M\#N$ of two topological manifolds is a topological manifold, however I don´t know how to prove that the connected sum of two differentiable manifolds is , in fact,…
user40276
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How to compute fundamental quantities in differential geometry for surfaces given in implicit form?

In my (undergraduate) readings in differential geometry, it seems that they assume that only parametrized surfaces are relevant. But what do we do when we get a surface in implicit form such as: $$f(x,y,z)=0 \tag{?}$$ In very simple cases such as…
Red Banana
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Do 1-forms return scalars or co-vectors?

I've been browsing questions regarding 1-forms and their difference with co-vectors, and I have stumbled upon what follows. @magma, here, said: 1-forms are simply the linear operators that take a vector and give out a number So, thinking of a…
user240612
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