# Tagged Questions

Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such ...

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### How do conformal maps affect curvature?

Let $(\overline{M}^{n+1}, \langle \cdot, \cdot \rangle)$ be a riemannian manifold with riemannian connection $\overline{\nabla}$ and consider $M^n \subset \overline{M}$ an orientable hypersurface with ...
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### Finding the chern classes of line bundles over projective space using homotopy classes of clutching functions

I have just started to learn about characteristic classes and before learning more about the ways to compute them it would be nice to compute some examples using tools I already know. I only started ...
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### Concrete examples and computations in differential geometry

I've been studying differential geometry by myself for some time now. I studied a fair amount of the basic general theory and gone through a lot of the exercises from several textbooks. Lately I ...
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### Solving a 2nd-order elliptic PDE with non-constant coefficients

I wonder how I can solve the following 2nd-order PDE on the positive semiplane $\{x>0\}$: $$(\partial_x^2+\frac{1}{x}\partial_y^2)\phi=\delta(x-x_0)\delta(y).$$ I notice that the l.h.s. is the ...
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### An imcomprehensible proof on Arnold's Lectures on PDE, contact diffeomorphism

In Lectures on Partial Differential Equations, Arnold gives a theorem in the chapter Huygens' Principle in the Theory of Wave Propagation: Theorem 1 (the theory of support functions). The manifold ...
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### Poincaré-Hopf theorem and its applications

I'm reading the basics of differential topology to try to understand the Poincaré-Hopf theorem, its proof and its applications. My plan is as follows: 1) Study transversality: its homotopy stability +...
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### sheaf of differential forms - tangent sheaf [Hartshorne]

I'm reading section 8 Differentials of chapter 2 in Hartshorne. It's is extremely hard to me to understand the nature of the definitions: module of relative differential forms - sheaf of relative ...
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### Do balls optimize the boundary area for a fixed volume?

I recently saw this question, asking if the circle is the planar figure which has the least perimeter given the area. As far as I know, this is a classical problem, and the answer is affirmative. I am ...
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### Differentiable Manifold Hausdorff, second countable

Why do we generally require that a differentiable manifold be Hausdorff and second countable? Is this universally accepted in the definition? My Professor only required the Hausdorff condition for ...
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### Isomorphism between spaces of sections.

Let $M$ be a compact manifold and let $E_i \xrightarrow{\Large \pi_i} M$, $i = 1, 2$, be two (real or complex) vector bundles of the same rank $k$ over $M$. Assume we have metrics $g_1, g_2$ on $E_1$, ...
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### Every vector bundle has a metric connection?

Let $(E,g)$ be a vector bundle with a metric over a manifold $M$. Does $(E,g)$ always admit a compatible (metric) connection? If so, are there examples where there exists only one such metric ...
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### $C^k$ one-parameter family of metrics

Consider a smooth Riemannian manifold $M$ and a $C^k$ one-parameter family of Riemannian metrics $g_t$ on $M$. Here $k$ could be any integer, $k$ could be infinity, when the one-parameter family $g_t$ ...
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### Is the de Rham complex a free (commutative?) differential graded algebra?

A differential graded algebra (dg-algebra) is a monoid object in the category of chain complexes with respect to the usual tensor product of complexes. A (graded) commutative dg-algebra is simply a ...
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### When is a vector field hamiltonian with respect to some symplectic form?

Given a vector field $v$ on a $2n$-dimensional manifold, how many symplectic forms are there on $M$ that make $v$ a hamiltonian vector field? Alternatively, take the set of all $(H,\omega)$ pairs, mod ...
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### Coordinate map from cotangent space is smooth

I know that for a given coordinate system of the tangent space $\partial_1,..,\partial_n$ the coordinate map is smooth, as the coordinate map $\pi_i$ of ${T_xM}$ is nothing but the composition of the ...
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In algebraic topology there are two canonical "advanced" textbooks that go quite far beyond the usual graduate courses. They are Switzer Algebraic Topology: Homology and Homotopy and Whitehead ...
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### Formal adjoint of curvature (Yang Mills)

Currently reading a paper on finding solutions to the Yang Mills equation $D^*\Omega=0$, where $\Omega$ is the curvature and $D^*$ is the formal adjoint of the exterior covariant derivative $D$. ...
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### Manifold, exist smooth nonnegative function with regular value at 0?

If $X$ is any manifold with boundary, then does there exist a smooth nonnegative function $f$ on $X$, with a regular value at $0$, such that $\partial X = f^{-1}(0)$?
Let $G\hookrightarrow P\xrightarrow{\pi}M$ be a principal $G$-bundle with right action $\cdot$ and suppose we are also given a left action $\rho: U\times P\rightarrow P$ of some group $U$ on $P$. ...
Suppose $C$ is a 3-form, and $G$ is a 4-form defined by $G = dC$. Also, $M_{11}$ is an 11-dimensional manifold (without a boundary), $W_{6}$ is a 6-dimensional submanifold of $M_{11}$ and \$D_{\epsilon}...