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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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Is every umbilic connected surface with 0 curvature cointained in a plane?

Is every umbilic connected surface $S$ with $0$ curvature cointained in a plane? I know that the answer is "yes" if we also suppose that the surface is orientable. The argument is sketched below: It's possible to prove that every connected umbilic…
user79594
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Equivalent definitions of partition of unity?

On Wikipedia a partition of unity is a collection of continuous maps $\varphi_i$ from a topological space $X$ into $\mathbb R$ such that for all $x$ (i) $\sum_i \varphi_i (x) = 1$ (ii) there is a neighbourhood $N_x$ such that only finitely many…
user167889
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On maximality of differentiable structure

In the book Foundations of Differentiable Manifolds and Lie Groups (by Warner) the author defines a differentiable manifold to be a pair $(M, F)$ where $F$ is a maximal atlas. Usually in the literature, I have seen $F$ to be any atlas. Why would an…
user167889
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How do I show that if there is a diffeomorphism between two smooth manifolds, the manifolds have the same dimension?

I think I've somehow got to bring tangent spaces into the picture, but how do I solve this?
adrija
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Why do we need an orientable surface for Gauss map?

I'm learning Differential Geometry recently with do Carmo's book. In the book, Gauss map is define as a differentiable map from an orientable surface $\mathcal{S}$ to $S^2$ in such a way that for every point on a $\mathcal{S}$ it maps to the unit…
Henry
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About the curvature of a curve

Let $\alpha : I \rightarrow R^2$ be a smooth curve. For each $t \in I$ consider $N(t)$ the normal unit vector at the point $\alpha(t)$. Fix $\lambda > 0 $ a constant and define the parallel curve $\gamma(t) = \alpha (t) + N(t) \lambda$. Someone can…
math student
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How do I complete this proof involving the Hodge dual and inner product?

Here are the preliminaries: let $M$ be an $n$-dimensional differentiable manifold equipped with a metric $g$. Define two $p$-forms $\alpha,\beta\in \Omega^p(M)$ and an $n$-form $\eta=\sqrt{|g|}dx^1\wedge\cdots\wedge dx^n$ (the canonical volume…
Ryan Unger
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Find a rigid motion to transform the curve

Say I have a curve $$r(t)=\left(t + \sqrt3\sin t\;,\;\; 2\cos t\;,\;\; \sqrt3t-\sin t\right)$$ I have discovered it is a helix and I want to reparameterize the curve in terms of the standard helix form (r(t)=(acost, asint, bt)). I need to find a…
erika
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Push Forward on product manifold.

Some words before the question. For two smooth manifolds $M$ and $P$ It is true that $T(M\times P)\simeq TM\times TP $ If I have local coordinates $\lambda$ on $M$ and $q$ on $P$ then ($\lambda$, $q$) are local coordinates on $M\times P$ (right?).…
AndresB
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Some problem similar to Dido's problem

The question is : "Let $A$ and $B$ be two fixed points in $\mathbb{R}^{2}$. Given $L>$ length of $AB$. Show that the curve $\alpha$ joining A and B, with length $L$, which together with AB forms a Jordan curve (i.e. a simple closed curve), bounds…
user134927
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Compatible intersecting coordinate patches must map to the same dimensional $\mathbb R^n$.

I have a problem where I have to show that for two intersecting open subsets $U$ and $V$ of a topological manifold, if we have two homeomorphisms $\phi : U \to \mathbb R^n$ and $\psi: V \to \mathbb R ^ m$, and $\psi \circ \phi ^{-1} \mid_{\phi(U\cap…
fhyve
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Do Taylor series of analytic vector field depend on metric tensor?

Let $\mathcal{M}$ be $n$-dimensional Riemannian manifold. And let $F$ be an analytic vector field on it. By definition this means that $$f^i(x(q)) = f^i \vert_{x(p)} + \frac{\partial f^{i}}{\partial x_{j_1}} \vert_{x(p)} x^{j_1} +…
Artem
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Geodesic deviation on a unit sphere

No response to this on Physics Stack Exchange, so I'm hoping for better luck here. My question is, can anyone tell me where I'm going wrong trying to use the equation of geodesic…
Peter4075
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tangent spaces of manifols

I have two manifolds $E_{n}=\{([0:x_{1}:x_{2}],[y_{1}: y_{2}])\in \mathbb{C}\mathbb{P}^{2}\times \mathbb{C}\mathbb{P}^{1}\}$ e $V_{n}=\{([x_{0}:0:x_{2}],[y_{1}: y_{2}])\in \mathbb{C}\mathbb{P}^{2}\times \mathbb{C}\mathbb{P}^{1}\}$ and I need to…
user166147
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Properties of Tangent Vector of a Differentiable Simple Closed Curve in 2D

I think of a theorem about a differentiable simple closed curve in 2D that I would like to prove. Here it is: Let $C$ be a differentiable, regular, simple, closed curve in $\mathbb{R}^2$ parametrized by $\gamma:[a,b]\rightarrow\mathbb{R}^2$. Let…
Joel
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