# 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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### the definition of canonical divergence in information geometry

The divergence defined in information geometry is global. But when it comes to the canonical divergence of dually flat space, it uses dual affine coordinate systems which is weird because this require ...
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### How are the extended de Rham differential and the covariant derivative related by Cartan's 2nd structural equation?

I am reading Prof. N.Poncin's notes on fibre bundles and connections. You can find it here:https://orbilu.uni.lu/bitstream/10993/14274/1/MM4-9November2011.pdf In $\it{Section \ 4.5.2}$ the author ...
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### Water flowing from a vessel with curved sides

Suppose a hole is drilled perpendicularly into the side of the beaker which is full to the brim with a fluid (say water). This will result in water spurting out, travelling in a parabolic trajectory ...
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### Existence of diffeomorphism so that $\int (\phi^* f) \omega = 0$

Can someone help me with this question? Is a qual exam question and I have no idea how to tackle it. Prove or disprove: Let $f\in \mathcal{C}^\infty(S^n)$ be a smooth function and $x_1, x_2\in S^n$ ...
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### Relation between tangent spaces of (un)stable manifolds in Morse theory

After asking this question about signs in the Morse complex, I realised that my confusion is really about how tangent spaces to different (un)stable manifolds are related. So suppose we have a Morse ...
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### Tangential component of normal vector parallel along curve iff curve is geodesic?

Exercise 6.3 (Millman & Parker, Elements of Differential Geometry). Let $$X_N = N - \langle N, n \rangle n$$ be the tangential component of the normal vector $N$ of a unit speed curve $\gamma$ ...
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### On the proof of that fixed point set of an involution is a submanifold

Let $M$ be a smooth manifold, and let $f:M\to M$ be a smooth involution (i.e. $f^2=\text{id}$). If we introduce a Riemannian metric on $M$ so that $f$ is isometry, we can prove easily that the fixed ...
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### cusps of bivariate polynomial equation

I plotted an implicit curve: $$f(x,y)=\sum_i^N a_i x^i y^{N-i}=0 \;\;\;a_i\in R$$ and found that the curve has sharp corners (cusps). (area where f>0 are painted red, otherwise green) If this is ...
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### $f:\mathbf{R}^n \to \mathbf{R}$'s derivative in each argument has the same sign everywhere. What is $f$'s shape?

We have a differentiable $f:\mathbf{R}^n \to \mathbf{R}$ with the property that each partial derivative has the same sign everywhere in its domain. Does this mean that the sublevel sets of $f$ (sets ...
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### Orientability of $m\times n$ matrices with rank $r$

I know that $$M_{m,n,r} = \{ A \in {\rm Mat}(m \times n,\Bbb R) \mid {\rm rank}(A)= r\}$$is a submanifold of $\Bbb R^{mn}$ of codimension $(m-r)(n-r)$. For example, we have that $M_{2, 3, 1}$ is non-...