# 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.

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### Showing the differential of a map $f:S\to\Bbb R^m$ is linear?

I've been asked an exercise where I have to prove that the differential, at a point, of a map defined on surface, is a linear map. I have this definition and then this lemma where they prove this in ...
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### Is the spherical coordinate system a vector space?

There are basis in spherical coordinate system but I think it does not satisfy vector axioms. Is calling them basis an abuse of notation?
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### principal symbol of a differential operator composed with its adjoint

Consider a linear differential operator with injective (principal) symbol from the space of sections of a (real) vector bundle to that of another. Suppose that the adjoint of this differential ...
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### Second fundamental form of isometric embedding

There is this well-known Weyl Embedding theorem which guarantees that a metric of positive Gaussian curvature can be realized as a convex surface in $R^3$, which is unique up to rigid motion in $R^3$. ...
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### Lie derivative of a differential form

I have a differential $1$-form $\omega = x\mathrm{d}x + x\mathrm{d}y$ and I need to find its Lie derivative along $X = (x+y)\partial_{x} - 2y\partial_{y}$. The first approach is by using Cartan ...
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### Hyperboloid model exponential map

I am working with the hyperboloid model of hyperbolic space and I am trying to understand its exponential map. However I do not see how this map is derived so I would appreciate either an explanation ...
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### Showing that $df_p\neq 0$ for all $p\in f^{-1}(0)$?

I'm trying to solve the following problem: Consider the function $f: S^2 \to \Bbb{R}$ given by $f(x,y,z)=x^{2023}+y^{2023}+z^{2023}$. Show that $df_p\neq 0$ for all $p\in f^{-1}(0)$. I'm thinking ...
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### Prove that Kahler form induced by Fubini-Study metric on $\Bbb{P}^n$ is the generator of the $\Bbb{Z}$ coefficient cohomology

I try to prove that the Kahler metric induced from the Fubini-Study metric on $\Bbb{P}^n$ is a generator of the cohomology $H^2(\Bbb{P}^n,\Bbb{Z})$ My attempt first by the Poincare duality we know ...
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### characteristic classes in odd dimensions? [closed]

So Stiefel-Whitney, Pontryagin, Chern classes are all for even dimensions. There seem to be no characteristic classes for odd dimensions. Does that mean in odd dimensions everything is trivial?
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### How to tell the tangent bundle of $S^2$ from the bundle $S^2\times$ $y-z$ plane?

Intuitively, I would like to say that the tangent bundle of $S^2$ and the bundle ($S^2\otimes \rm{y-z\ plane}$) is different. By the latter I mean a product bundle, embedding in $R^3$ it is equivalent ...
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### How to derive $F^{\mu\nu}$ from $F_{\mu\nu}$?
Let $\vec A = [\phi, A_x, A_y, A_z]$ be a vector field, and a covariant tensor field be defined as $$F_{ab} = \frac{\partial A_a}{\partial x_b} - \frac{\partial A_b}{\partial x_a}$$ Can the ...
### Does $\partial\bar{\partial}+\bar{\partial}\partial=0$ imply integrability of the almost complex structure?
Let $X$ be an almost complex manifold. A well-known result says that $\bar{\partial}^{2}=0$ implies the integrability of the almost complex structure. My question is what about \$\partial\bar{\partial}+...