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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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What is a diffeomorphism?

I'm looking for a simple (i.e. as just a rough outline with little differential geometry) definition/ explanation of what a diffeomorphism is. I tried reading the Wiki page but it made no sense to me as a physicist. To give some context, I'm…
user16307
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Expressing Differential Form in Different Coordinates

All: Please forgive me, I'm new and my editing/Latex needs improvement. I'm trying to derive the formula for change of variables for the differential form $\omega=dx\wedge dy$ in standard $xy$-coordinates in $\mathbb R^2 $, into polar coordinates in…
Confused
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Tangent bundle of $S^2$ not diffeomorphic to $S^2\times \mathbb{R}^2$

I am trying to show that the tangent bundle of $S^2$ not diffeomorphic to $S^2\times \mathbb{R}^2$. This is from an exam, where there is a hint stating that this is more than showing that $TS^2$ is non-trivial. I know how to show the hairy ball…
user54631
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Is the chart function of a smooth manifold a differomorphism, not just a homeomorphism

It's clear that a smooth chart on a manifold is a diffeomorphism. To me, the fact that smoothness of a manifold implies the smoothness of the transition function between the representation of two charts (whose domains overlap in M) should also…
luysii
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What's special about $C^\infty$ functions?

In my experience, people usually use "smooth" to mean "as smooth as I need for the upcoming proofs." Those who want to be more formal might insist on smooth meaning $C^\infty$. While the operator taking $f$ to its Taylor series at some point in its…
Kevin Carlson
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What type of object is a differential form?

This is a naive question; apologies in advance. For a point $p \in M$ on a smooth manifold $M$, a differential form can be viewed as a map $$T_p M\times \cdots \times T_p M \to \mathbb{R} \;.$$ What puzzles me about this object is that it is not…
Joseph O'Rourke
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Is the group of diffeomorphisms a Lie Group?

consider a smooth manifold and the group of diffeomorphisms (or (local) isometries in case of riemannian manifolds) $\varphi:M \rightarrow M$. How can one define a smooth structure on this group, s.t. it becomes a Lie group? Regards
Braten
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divergence of a vector field on a manifold

I've been asked to show the following: For a vector field $V$ on a semi-Riemannian manifold with metric $g$ that $$Div \cdot V = \frac{1}{\sqrt{\det(g)}}\partial_i\left(\sqrt{\det(g)}V^i\right)$$ I know we're supposed to use Christoffel symbols as…
Misha
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Prove a Levi-Civita connection gives $\nabla_XY(p)=\partial_t|_{t_0}[P^{-1}_{c_0,t_0,t}(Y(c(t)))]$ with $P$ parallel transport

I'm having trouble with the following exercise in do Carmo's Riemannian geometry. Let $X$ and $Y$ be differentiable vector fields on a Riemannian manifold $M$. Let $p \in M$ and let $c: I \to M$ be an integral curve of $X$ through $p$, i.e. $c(t_0)…
Sam
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Is the set of singular matrices ever a differentiable manifold?

I can see that invertible matrices are a differentiable manifold however I don't know how to show that something is not a differentiable manifold so easily. Is it ever the case that singular matrices form a differentiable manifold?
Ben Carter
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Different notions of isometry for Riemannian $2$-manifolds

There are two notions of isometry between Riemannian $2$-manifolds: a distance-preserving map $f$ with $d(x,y) = d(f(x),f(y))$ and a "metric-preserving" map $f$ with $I(x) = I(f(x))$ ($I(x)$ being the first fundamental form) The second isometry…
Hans-Peter Stricker
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manifold structure on on a finite dimensional real vector space

I am reading Warner's Differentiable Manifolds I do not get one example which is Let $V$ be a finite dimensional real vector space. Then $V$ has a natural manifold structure. If $\{e_i\}$ is a basis then the elements of the dual basis $\{r_i\}$…
Myshkin
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Calculating the integral curves of a vector field

How do I caluclate the integral curves of a vector field, i.e. how would I go about calculating the integral curves of: Define the vector field in $\mathbb{R}^3$ by: $ u = x_1\displaystyle\frac{\partial}{\partial x_2} +x_2\frac{\partial}{\partial…
hmmmm
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Chain rule of differential in smooth manifold

I am having trouble understanding the chain rule in smooth manifolds (unfortunately that part of the book is an exercise). I see a resemblance to the chain rule in $\mathbb{R}^n$ but do not understand from the definition of the differential…
Emil
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Want to learn differential geometry and want the sheaf perspective

I would like to learn some differential geometry: basically manifolds, differentiable manifolds, smooth manifolds, De Rham cohomology and everything else that is pretty much part of a course in differential geometry. I do however know some deal of…
user7713
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