# 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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### Confusion with conclusion to positive mass theorem

I am trying to understand the positive mass theorem as it is presented in the survey paper by Corvino and Pollack http://arxiv.org/abs/1102.5050 I am fundamentally confused by the structure of their ...
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### Convex, closed plane curve is a Jordan curve

The claim I'd like to prove or disprove is in the title. Here, convexity means every tangent line at any point on the curve has the whole curve in one half of it. The curve being closed means that it'...
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### What are “generalized bases” really called, and where can I learn more?

(Notation: $f \diamond g$ means the composite $g \circ f$.) The following situation occurs frequently: We have an $\mathbb{R}$-algebra $A$, together with a distinguished set $I$ (the "indexing set"),...
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### Are there characteristic classes for symplectic vector bundles?

Given a real vector bundle, say the tangent bundle of a manifold, some obstructions to this being the underlying real bundle of a complex bundle come from characteristic classes. These can prove the ...
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### Derivative of the integral of a pull-back form

Let $\omega$ be a $n$-form on the smooth compact manifold $M$ without boundary. Let $X$ be a smooth vector field on $M$ and $\phi_t$ the associated flow. Let $A(t)=\int_M \phi_t^* \omega$. How can we ...
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### Orientability of manifold via covering spaces

Let $f\colon M\rightarrow N$ be a regular covering map between connected differentiable manifolds $M,N$ with $M$ orientable. Prove that $N$ is orientable if and only if every deck transformation ...
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### conformal structure of a disc

I wonder if the conformal structure of the unit disc $D^2=\{(x,y):x^2+y^2\leq 1\}$ is unique. More precisely, given a Riemannian metric $g$ on $D^2$, is it always true that $g=e^{2u}g_0$, where $g_0$...
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### Does an unbounded gradient implies a non-vanishing hessian determinant

Let $\Phi:\mathbb{R^{n-1}\to R}$ be a vector function. Suppose that exists a set $S_1\subset \mathbb R^{n-1}$ on which $G=\nabla\Phi$, the gradient of the function is unbounded. Does that imply ...
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### Fundamental group and curvature

Is there any paper about the $\pi_1$ group and curvature ? Because how close a curve depends on the curvature near the curve . I think there must have some condition which decide whether there is ...
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### Flow of a vector field, equivalent characterizations

Given a vector field $X$ on the manifold $M$ its flow is, loosely speaking, a map $F= F(t,x)= F^t(x)$, in the variables $(t,x)\in I\times M$, such that the curve $t\mapsto F^t(x_0)$ is the unique ...
### The Lie Algebra of $O(n)$ is the set of $n \times n$ skew-symmetric matrices
I'm trying to show that the Lie Algebra for $O(n)$ is the set of $n \times n$ skew-symmetric matrices. Here is what I have so far. Since $O(n)$ is the union of two disjoint subsets, the matrices with ...