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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62 views

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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94 views

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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42 views

Proper map and manifolds

Let $M$ and $N$ two manifolds which have the same dimension, $f:M\to N$ a map $\mathcal{C^\infty}$. We suppose that $M$ is compact and we have $b$ a regular value of $f$. First, I have to prove that $...
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Show that a smooth disk is a free homotopy of boundary curve to constant

Problem says: A smooth disk $\mathscr{D}$ in a surface is the image of the unit disk $x^{2}+y^{2}\leq1$ in $\mathbb{R}^{2}$ under a one-to-one regular map $F$ . Show that the 2-segment ...
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29 views

There is no smooth map from $R^3$ to $R$ such that there i

I have to prove there is no smooth map $f:\mathbb{R}^3 \to \mathbb{R}$ and no regular value $y$ of $f$ such that $f^{-1}(y)$ is the projective space of dimension 2. From the pre-image theorem, $f^{-...
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68 views

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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1answer
26 views

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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88 views

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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11 views

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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49 views

$f:\mathbf{R}^n \to \mathbf{R}$'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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1answer
51 views

Normal variation of embedded surfaces

Let $M^3$ be a complete riemannian manifold and $\Sigma ^2\subset M^3$ a embedded minimal compact surface. Consider the normal variation $\phi: \Sigma \times \Bbb{R}\to M$ given by $$\phi(p,t)=\exp_p(...
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1answer
39 views

determine the slope of a point on a ellipse

the equation of ellipse is $Ax^2 + By^2 + Cx + Dy + Exy + F = 0$ for slope, $2Ax+2By*dy/dx+C+D*dy/dx+Ex*dy/dx+Ey=0$ so, $(2By+D+Ex)*dy/dx=-(2Ax+C+Ey)$ => $dy/dx=-(2Ax+C+Ey)/(2By+D+Ex)$ This ...
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19 views

Boundedness of scalar curvature gives boundedness of sectional curvature?

Let $(X,\omega)$ be a compact Kahler manifold which the scalar curvature is bounded then the sectional curvature is bounded?
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60 views

Integral of an interior product is 0

Here is a question about problem 3.12 from Do Carmo's Riemannian Geometry. $M$ is a compact orientable and connected Riemannian $n$ manifold. $f$ is a differentiable function on $M$ such that $\Delta ...
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26 views

For a minimal surface $M$ under Mean Curvature Flow, can it evolve between minimal surfaces continually?

I don't know much about this subject at all; I'm only just getting into it. As it turns out, a physicist friend of mine asked me a formulation of the following: Suppose $M$ is a surface in $\mathbb{R}...
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1answer
22 views

Partial derivative of an inner product and a linear transformation

Let $u(x)$ be a self adjoint operator on $R^n$, and $\langle\underline{\ },\underline{\ }\rangle$ the usual dot product. I have to show $f(x)=\langle x,u(x)\rangle$ is differentiable over all of $R^...
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1answer
27 views

Poisson point processes on Riemannian manifolds

Most treatments of Poisson point processes I have encountered define them on $\mathbb{R}^d$, with the key property that the number of points in a bounded region $R$ will have a Poisson distribution ...
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1answer
20 views

Trace of Ricci flow equation

The Ricci flow equation as is known is given as: $\partial_t g_{ij} = -2R_{ij}$. If I take the trace/contract the indices of both sides, does this imply that: $\partial_t g = -2R$, where $g$ is ...
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41 views

Book on differential Geometry with application to General Relativity

Does anybody know of a good book on differential geometry that has applications to general relativity and also focuses on geometrical intuition? I need a book that is not as rigorous as one that is ...
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1answer
98 views

Is $(\mathbb R^3\setminus \{0\})/\mathbb R^*$ a smooth manifold?

Let $G=\mathbb R^*$ act on $X=\mathbb R^3\setminus\{0\}$ by pointwise multiplication. That is for any $t\in\mathbb G$ and $(x_1,x_2,x_3)\in X$ we have $$t\cdot(x_1,x_2,x_3)=(tx_1,tx_2,tx_3)$$ Is ...
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1answer
45 views

Visual explanation of the indices of the Riemann curvature tensor

I'm trying to understand the meaning of the Riemann curvature tensor, but I don't seem to be ready yet to understand the detailed rigorous definition. Anyway, I managed to understand (this gif was ...
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25 views

Volume element induced by metric tensor

I am using Wald's General Relativity. In Appendix B he says that given a smooth Lorentzian manifold $(M, g)$ then there a natural choice of volume element $\epsilon$ specified up to sign by $\epsilon^{...
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31 views

diffeomorphism and product of open sets

Let $D$ be an open set in $\mathbb{R}^2$. Is $D\times\mathbb{R}^2$ always diffeomorphic to $\mathbb{R}^4$ ? I think this is not true intuitively, but I can't find the counterexample.
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47 views

Stokes' Theorem and path integrals in $\mathbb{C}$

I have seen very short proofs of the Lemma of Goursat and Cauchy's Theorem using Stokes' Theorem. I have learnt Stokes' Theorem in the setting of (not complex) smooth manifolds as described in https://...
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42 views

Radius of $\mathbb{CP}^n$

The question I am asking is basically Is it possible/usual to define a "radius" for certain metrics on $\mathbb{CP}^n$ in analogy to the case for $S^n$? To provide some more context about what I ...
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1answer
57 views

Explicit verification of signs in Morse complex

I'm trying to check by hand that the signs in the Morse complex, defined via choices of orientations on the unstable manifolds, lead to $\partial^2=0$. The books I've looked in seem to say either ...
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1answer
31 views

Induced metric on a one-sheet hyperboloid

I am trying to find the induced metric on a one-sheet hyperboloid. Suppose we use cylindrical coordinates $(r, \theta, z)$ for the ambient space in which the hyperboloid is embedded. The hyperboloid ...
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When can a vector field be “lifted” to a spinor field with preservation of continuity?

Suppose we are given a vector field $\xi ^a (x)$ on some region of Minkowski spacetime which is null everywhere, $$\xi^a(x) \xi_a (x) =0.$$ For every point of our region we can choose a spinor $\...
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25 views

Real projective space, tangent space, orientation

In the O'neill's differential geometry text, there are following problems. The projective plane $P$ is defined as follows. As you know, this $P$ is an abstract surface. So, we have to define the ...
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38 views

$1$-forms $\omega$ on $S^1$ are the differential of functions provided $\int_{S_1}\omega=0$

Prove that a (smooth) $1$-form $\omega$ on $S_1$ such that $\int_{S_1}\omega=0$ is the differential of some $f:S^1\to\mathbb{R}$. Hint: Let $h:\mathbb{R}\to S^1$ defined by $h(t)=(\cos(t),\sin(t))...
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23 views

Contraction of a tensor by vector

If I have an expression: $g(X,Y)+s(Y) =0$, and I want to contract it by Y, that is to write it as $X+?$, what can I do with s(Y)? Here, g is a metric and s is a (1,0) tensor.
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1answer
46 views

Restriction of an $n$-form to a submanifold

I am currently studying exterior calculus on manifolds and am not sure if I understand things correctly. In my textbook (R.W.R. Darling, Differential Forms and Connections) there is an example of a $2$...
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1answer
59 views

Can the nabla symbol be used for a covariant derivative?

Can $$ \nabla_{\nu} A^{ \mu\nu} $$ represent a covariant derivative with respect to $\nu$? If not what can it be? I'm reading a textbook on General Relativity, and such operations appear without any ...
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3answers
51 views

Parametrization of the sphere and the torus.

Is there a way to find easily the parametrization of the sphere and the tore ? I see on wikipedia that for the sphere it's $(x,y,z)=(\sin \theta\cos \varphi,\sin\theta\sin\varphi,\cos\varphi)$ with $\...
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17 views

Gauss curvature and bisectional curvature

Let $f:\mathbb CP^1\to X$ be an smooth embedding which its image is the curve $C$ where X is a Kahler manifold. Then is it correct that $$\sup_{\mathbb CP^1}K(f^*\omega)<\sup_C K(\omega)$$ where ...
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1answer
43 views

What am I doing wrong when calculating this pullback?

Let $\omega = \sum_{j=1}^{n+1} x_j dy_j - y_j dx_j $ be a differential form on the sphere $S^{2n +1}$. Let $G = Z_2$ be the group acting on the sphere. I want to apply the following proposition to ...
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32 views

Orientability of differantiable manifold of orthogonal matrices

I want to find out if differentiable manifold of matrices $M=\{A_{(3\times3)}(\mathbb{R}): A^TA=16E\}\subset\mathbb{R}^{3\times3}=\mathbb{R}^9$ is orientable. It is only worth proving that orthogonal ...
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0answers
47 views

Why say “exists smooth structure” instead of “is a smooth manifold”?

As some of you know I started to learn differential geometry some weeks ago. Now I came across this theorem (you can find it in Lee's Intro to Smooth Manifolds but it is in many other books also: ...
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27 views

When is a stable domain in a minimal surface area minimizing?

A stable domain $D$ in a minimal surface $S\subset \mathbb{R}^3$ is a domain for which the area-functional $A(t):=\int_{S_t}dS_t$ has non-negative second derivative, i.e. $A''(0)\geq 0$, for all ...
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1answer
56 views

Exterior derivative of complex differential form

I have this question, from several complex variables: Start with the differential form: $$\omega(z)=\sum_{\nu=1}^{n} \frac{(-1)^{\nu-1}\bar{z}_{\nu}}{|z|^{2n}} d\bar{z}[\nu] \wedge dz, $$ where $dz=...
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1answer
38 views

Demonstration of a basic formula involving differential forms

I'm writing some notes on Lie Groups and I'm not sure if I should demonstrate this formula or not. Assume $\omega$ is a differencial form and $X,Y$ fields con a Manifold M, is there a simple way to ...
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Finding braches of equilibria

Consider the system of two equations in three variables $(x,y,z)$: $$\left\{ \begin{array}{rl} x+y+z+x^7+y^7+z^9 &= 0\\ x-y+z+1-\cos z &=0 \end{array}\right.$$ The point $x^* = (0,0,0)^...
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2answers
51 views

Orientability of surfaces

How to prove that a surface is orientable? Is it true that the union of two orientable surfaces is orientable? How to prove that? For example, is the union of the hemisphere $$z = \sqrt{1 - x^2 - y^2}...
27
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2answers
247 views

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-...
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1answer
50 views

Solving the Euler-Lagrange equations for geodesics

I am trying to find geodesics on the following metric: $ds^2 = dx^2 + x^2 dy^2$ Setting $dx \rightarrow \dot{x}, dy \rightarrow \dot{y}$ in $ds^2$ i get following Lagrangian: $L = \dot{x}^2 + x^2 \...
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1answer
25 views

Christoffel Symbols - Showing Equality

Using: $$\sum_{j} g_{ij}g^{jk}=\delta_i^k$$ $$\Gamma_{ijk}+\Gamma_{jik}=\frac{\delta g_{ij}}{\delta u^k}$$ $$\Gamma_i{^j}_k = \Gamma_k{^j}_i = \sum_{l} g^{jl}\Gamma_{ilk}$$ Show that: $$g_{11}\frac{\...
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3answers
55 views

If $\gamma :[a,b]\rightarrow \mathbb{R}^3$ is smooth then $\gamma(t)=x$ has finite number of solutions

Let $\gamma :[a,b]\rightarrow \mathbb{R}^3$ be a smooth curve ($\gamma$ is differentiable with $\gamma'(t)\neq \mathbf{0}$ for all $t\in[a,b]$). Show that, for $x\in\mathbb{R}^3$, the equation $\gamma(...
2
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1answer
60 views

Frobenius theorem

I came across the following conclusion in a textbook, but can't really understand it. I would be grateful if anyone could elaborate: Assume that we have two linearly independent vector fields $V_{1},...