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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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Gauss-Bonnet-Chern theorem

Good morning/day/evening/night, I was presented to the generalized Gauss-Bonnet-Chern theorem for hypersurfaces in Euclidean space; For a closed, even dimensional manifold $M$ with dimension $n$ embedded in $\mathbb{R}^{n+1}$ we have $$\int_M…
M.B.
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Volume form on $S^2$

I have some basic understanding problem on this. Any help is appreciated: The volume form of $S^2 \subset R^3$ is given by $$ \omega = x \ dy \land dz-y \ dx \land dz +z \ dx \land dy$$ In polar coordinates this becomes $$ \omega = sin\Theta\…
alexl
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A curve where all tangent lines are concurrent must be straight line

I'm trying to solve this question in the classical Do Carmo's differential geometry book (page 23): A regular parametrized curve $\alpha$ has the property that all its tangent lines pass through a fixed point. Prove that the trace of $\alpha$ is a…
user42912
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Push forward or differential : is there a link with the differential of a function?

My question is really naive but in differential geometry we also call differential the push-forward associated to a function $F : M \rightarrow N$ between two manifolds $M$ and $N$. But I don't see the link between this map $F_*$ and the usual…
StarBucK
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Area of Mobius strip

I want that to give a meaning to the notion of area for Mobius strip. I know that Mobius band is nonorientable surface. How can I set up an integral to compute it? What's your idea for the following formula? $\boldsymbol…
Bimanifold
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Structures on torus

Quotienting $\mathbb R^2$ by different lattices isomorphic to $\mathbb Z^2$, we get different tori. Somehow I think of the tori as having different "structures", but thinking more about it, I am not quite sure what different structures I am really…
Earthliŋ
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Covariant derivative

In my book the author makes the remark: If $X,Y$ are smooth vector fields, and $\nabla$ is a connection, then $\nabla_X Y(p)$ depends on the Value of $X(p)$ and the value of $Y$ along a curve, tangent to $X(p)$. When I got it right, then we can…
Braten
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The set of all critical points of a smooth map is closed

Let $f : \mathbb{R}^m \to \mathbb{R}^n$ be a smooth map. How do I show that the set of all critical points of $f$ is closed in $\mathbb{R}^m$? (Here, a critical point is a point $x \in \mathbb{R}^m$ for which the derivative $Df_x : \mathbb{R}^m \to…
Sayantan
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A smooth function $f:S^1\times S^1\to \mathbb R$ must have more than two critical points.

I am trying to show that a smooth function $f:S^1\times S^1\to \mathbb R$ must have more than two critical points. Since $f$ attains maximum and minimum, it must have at least two critical points. How would one show that they can't be two? If one…
Dimitris
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defining smooth functions on smooth manifolds

The standard approach to defining smooth functions $f:M\to\mathbb{R}$ on a topological manifold $M$ equipped with a smooth structure (i.e., a maximal smooth atlas) $\mathcal{A}$ is the following. Say a function $f$ is smooth at $p\in M$ if there…
symplectomorphic
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Are all metric tensors diagonal?

If I understand correctly, one way to get the components of a metric tensor (treating it like a matrix here) is to look at the $ds$ interval. Isn't that interval always in terms of sums of $dr^2+d\theta^2$ etc, meaning that the metric tensor will…
user87611
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Gradient in differential geometry

I am a graduate student in physics trying to learn differential geometry on my own, out of a book written by Fecko. He defines the gradient of a function as: $ \nabla f = \sharp_g df = g^{-1}(df, \cdot ) $ This makes enough sense to me. However,…
Christopher Reilly
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how does parallel transport work on the sphere

I am starting to try to understand the concept of parallel transport from differential geometry and I have run into a problem. I have been attempting to compute parallel transport on a sphere (embedded into $\mathbb{R}^3$, with the Levi-Civita…
Jakub Konieczny
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Christoffel symbols vanish in a system of normal coordinates.

I'm reviewing for a differential geometry exam and am getting stuck in a proof. This is based on question 4 from section 4-6 from little Do Carmo. Show that in a system of normal coordinates centered at $p$, all the Christoffel symbols are zero at…
doppz
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What results explicitly require second countability or the Hausdorff condition for a manifold?

It seems like the "locally Euclidean" condition in the definition of a manifold is most important and most frequently used. I've also seen the standard examples of spaces which are locally Euclidean, but not second countable or Hausdorff. I'm…
user2942
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