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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Conditions on characteristic polynomial to define a matrix submanifold.

I'm trying to find conditions on the characteristic polynomial, $p$, of a matrix such that the pre-image of matrices with characteristic polynomial $p$ form a manifold. More precisely, we can write ...
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1answer
32 views

The matrix square root is not differentiable on the boundary of the manifold of positive semi-definite matrices?

$\newcommand{\psym}{\operatorname{P}_{\ge 0}}$ $\newcommand{\Sig }{\Sigma}$ Let $\psym$ denote the subset of symmetric positive semi-definite matrices. Let $S:\psym \setminus \{0\} \to \psym ...
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2answers
42 views

Does immersion depend on the selection of chart

Let $F: N \rightarrow M $ be a smooth map from smooth manifold $N$ of dimension n to smooth manifold $M$ of dimension m. $F$ is an immersion at $p \in N $ if $F_{*,p}$ is injective. However $F_{*,p}$ ...
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Christoffel Symbols and the change of transformation law.

I have seen it written that the change of co-ordinate form is given by the following: $$ \tilde \Gamma^{i}_{jk} = {\partial \tilde x^i \over \partial x^\alpha} \left [ \Gamma^\alpha_{\beta ...
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0answers
13 views

definition of covariant derivative (along curve)

An affine connection on a smooth manifold $M$ is a map $\nabla: \mathcal{V}(M) \times \mathcal{V}(M) \to \mathcal{V}(M)$ satisfying several properties, where $\mathcal{V}(M)$ denotes the set of smooth ...
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1answer
19 views

Every open cover of a smooth Manifold has a regular refinement

I am trying to understand the proof of Let M be a smooth manifold. Every open cover of M has a regular refinement. The proof begins as follows [Lee] : Let $X$ ...
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1answer
30 views

How to calculate this derivative in differential geometry

Given a symmetric matrix $A$ and a function from generalized linear group to generalized linear group $$f: \text{GL}(n,\mathbb{R})\rightarrow \text{GL}(n,\mathbb{R}), g\mapsto g^TAg$$ For $\forall ...
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2answers
82 views

Example of a sum of complete vector fields

Can anyone give an example of two vector fields $X_1$ and $X_2$ which are complete but their sum $X_1+X_2$ is not complete?
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294 views

Horizontal tangent line of a parametric curve

Suppose $x=t^2,y=t^3$ is a parametric curve. Here's a quote from my textbook: The origin, which corresponds to $t=0$, is a singular point of the parametric curve, because $dx/dt=2t,dy/dt=3t^2$ are ...
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29 views

How these charts are written?

The spherical coordinate map$$σ(u, v) = (\cos u \cos v, \cos u \sin v,\sin u), −π/2 < u < π/2, −π < v < π,$$ and its variation $$σ˜(u, v) = (\cos u \cos v,\sin u, \cos u \sin v), −π/2 < ...
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1answer
12 views

Is there exist a smooth function $f:\mathbb{R}^{m}\to M$ such that $f|_{U′}=\phi^{-1}$?

Let $M$ topological smooth manifold and $(U,\phi)$ chart fixed with $\phi(U)=U′$ open in $\mathbb{R}^{m}$. Is there exist a smooth function $f:\mathbb{R}^{m}\to M$ such that $f|_{U′}=\phi^{-1}$? I ...
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1answer
15 views

Orthonormal Frame as a function

Let $M$ be a smooth manifold. We know that the frame at a point $p\in M$ can be defined as an isomorphism $f:\mathbb{R}^n\longrightarrow T_pM$. Is there a way of defining an orthonormal frame in a ...
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0answers
29 views

Definition of “differential” [duplicate]

I am confused about the definition of "differential". Sometimes I see it is a pushforward mapping $df:TM\rightarrow TN$, which gives another tangent vector when acting on a tangent vector. ...
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8 views

Given edge of regression find the characteristic lines of planes

A single $\infty^1 $ of planes moves forming its envelope as a tangential developable,further producing the circle $$ x^2 + y^2 = a^2 $$ as its edge of regression as a sharp cuspidal edge. Please ...
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1answer
50 views

Is the hemisphere of $S^4$ the unique compact 4-dimensional manifold with $\partial K = S^3$?

The first question is Is the hemisphere of $S^4$ the unique compact 4-dimensional manifold with $\partial K = S^3$? In other words, is it obvious? The question stems from a theoretical physics ...
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0answers
13 views

Find a differentiable curve on the paraboloid $\,z= 2x^2 + y^2$ with minimum curvature

Let $S$ be the graph of the function $\,f(x,y) = 2x^2+ y^2$ in $\,\mathbb{R}^3$ (a paraboloid with vertex at the origin). It is clear that $S$ is a regular surface which can be parametrized by the ...
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43 views

Is there a connection between topological mixing and squashing functions used in neural networks?

Sigmoid, ReLU, tanh, logistic -type "squashing" functions are popular in neural networks to introduce nonlinearity into the transformations of the input vector, allowing the network to fit complex ...
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26 views
+50

Can you determine the length of a curve by the lengths of its projections onto planes?

If $\Gamma \subset \mathbb R^n$ is 1-rectifiable, then its Hausdorff measure is equal to its integralgeometric measure. That is, $$\mathcal H^1(\Gamma) = \int\limits_{G(1,\mathbb R^n)} \int\limits_K ...
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1answer
21 views

Reference about the space of closed curves in Riemann manifold

Some days ago, I listened a report about the width of Riemann manifold. I am interesting in the space of closed curves in Riemann manifold. It seemly has good topology and different construction.For ...
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0answers
15 views

Show that the convex neighborhood in a Riemannian Manifold are subset contractibles

Show that the convex neighborhood in a Riemannian Manifold are subset contractibles (to any of their points). A subset $A$ of the differential manifold $M$ is contractible to the point $a\in A$ ...
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18 views

Equation of the geodesic corresponding to the metric $ds^2=du^2+f^2(u)\,dv^2$

I have the surface of revolution $\sigma(u,v)=(f(u)\cos v,f(u) \sin v,g(u))$ with $f > 0$ where the profile curve $u \to(f,g)$ has unit speed. I know the metric associated with the surface patch ...
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1answer
9 views

Show $\gamma(I)$ is a regular parametrized curve in $S$

Let $U={(u,v) \in \mathbb{R}^2 | u>0,v>0}$ be the open set in $\mathbb{R}^2$ and let $\sigma:U\to\mathbb{R}^3$, $\sigma(u,v)=(u^2,2v,1/v)$ I've shown that $S=\sigma(U)$ is a parametrized ...
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1answer
44 views

Finding A 1-form on $R^2 - {\{(0,0)}\}$

I want to find a 1-form on $R^2 - {\{(0,0)}\}$ such that $w(Y) = 0$ and $w(X) = 1$. Here, $$X = -y\frac{\partial }{\partial x} + x\frac{\partial}{\partial y}\ \text{and}\ Y = x\frac{\partial ...
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0answers
5 views

maximal volume/diameter of a set of simplexes

I am trying to develop a simplicial integral in $R^n$ and I am faced with the problem of controlling the "compacity" of a set of simplexes: Let $S$ be a finite set of n-d simplexes in $R^n$. Define ...
2
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1answer
37 views

What value of $c$ makes this Riemannian metric complete?

I was given the following question in my differential geometry class. The instructor does not use a textbook, and gives only theorems and proofs with no examples, so I don't know how to do ...
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0answers
41 views

Explicit form of the exponential map

I am stuck by the following problem. Let $ g = 1/y^2 (dx \otimes dx + dy \otimes dy)$ be a Riemannian metric on the half-plane $ S = \mathbb{R}^2_{y > 0}$. What is the explicit form of the ...
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0answers
22 views

Software of symbolic computation

In Riemann geometry, there are many complex compute , for example in the picture below.If want to get 2.5.16 it needs about 3 page to compute. And it is easy to mistake because it is complex. But the ...
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0answers
21 views

Viewing geodesic polar coordinate generated parallel circles as intersecting points of involutes.

Geodesic polar coordinates on surfaces studied first by Leibnitz (and later by Gauss) referred to them (parallel circles, "concentric" ovals) as involutes. Does it mean that length of tangent from ...
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1answer
453 views

Surface integral for a scalar function defined on a discrete surface

Imagine a polyhedral, discrete surface embedded in $\mathbb{R}^3$. Its faces are all triangles. For each vertex, one can compute the discrete mean and Gaussian curvatures and evaluate the sum of ...
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71 views

Why is a PDE a submanifold (and not just a subset)?

I struggle a bit with understanding the idea behind the definition of a PDE on a fibred manifold. Let $\pi: E \to M$ be a smooth locally trivial fibre bundle. In Gromovs words a partial differential ...
2
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1answer
46 views

A necessary and sufficient condition for the admittance of integrating factor

Let $\omega$ be a smooth 1-form on a smooth manifold $M$. A smooth positive function $\mu$ on some open subset $U\subset M$ is called an integrating factor for $\omega$ if $\mu\omega$ is exact on ...
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10 views

Asymptotic lines of Dini surface

Is there a reference for parametrization of twisted pseudosphere lines of curvature and asymptotics? How does one proceed after obtaining coefficients of second fundamental form? EDIT1: In standard ...
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0answers
29 views

Smoothing a continuous section in a fibre bundle.

Let $\pi: X \rightarrow M$ be a smooth fibre bundle and let $p^{1}_{0} : X^{(1)} \rightarrow X$ be its 1-jet bundle. Suppose there is a $\mathcal{C}^{1}$ section $h: M \rightarrow X$ such that it is ...
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0answers
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How to show one-one and onto for a function of homogeneous coordinate [on hold]

Let M = ([0, π] × R)/ ∼ where we identify the points (0, t) ∼ (π, −t) for all t ∈ R (i.e. M is the infinite Möbius strip). Let f : M → (RP 2 r {(0 : 0 : 1)}) be the map defined by f([(θ, t)]) = (cos θ ...
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1answer
31 views

Basic question about the metric tensor

So I am reading Barrett O'Neil's Elementary Differential Geometry, which defines the metric tensor $g_p$ on a surface $M$ to be a bilinear symmetric positive-definite function on the set of ordered ...
2
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1answer
37 views

maximal volume/diameter of polyhedron

I am trying to develop an integral in $R^n$ and I am faced with the following problem: Given a polyhedron $P$ in $R^n$ of diameter d, define the "compactness" of $P$ as the quotient of the volume of ...
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1answer
10 views

Relation between derivatives of chart, derivatives of unit normal and Gaussian Curvature

Don't know how the prove this apparently simple relation: if $x(u,v)$ is a chart of a surface $S$, with unit normal $N(u,v)$, then $N_u\times N_v=Kx_u\times x_v$, where $K$ is the Gaussian curvature. ...
2
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1answer
22 views

How transversality condition implies that a value is regular?

Currently I am self-learning some manifold theory and just come across concept of functions transverse to submanifolds. It seems that this concept is used a lot for proving regularity of values, but I ...
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0answers
21 views

On lifts of a trajectory of a quadratic differential

Let $X$ be a Riemann surface of genus $g\ge 2$ and $q$ a holomorphic quadratic differential on $X$. The differential $q$ defines a flat metric with conical singularities on $X$: if $q=f(z)dz^2$ ...
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1answer
20 views

Oriented atlas on a circle

I'm trying to find an oriented atlas on the circle $S^1$, i.e., I want to find an atlas for $S^1$ such that for any two overlapping charts $(U,s)$ and $(V,t)$ of the atlas, the derivative $d s/d ...
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0answers
25 views

Show that $(x\land y)z + (y\land z)x + (z\land x)y=0.$ where $x\land y=(x \times y)\cdot N$.

Let $P\subset \mathbb{R}^3$ be a plane through the origin and $N$ be a unit normal to $P$. For $x,y \in P$, set $x\land y=(x \times y)\cdot N$. Then for any three vectors $x,y,z \in P$, we have ...
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1answer
21 views

Integration on k-1 form

If $\omega$ is a $k-1$ form on a closed $k$-dimensional manifold $M$ then $\int_M d \omega = 0$. I'm looking for a short proof to this problem, would Stokes be helpful?
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28 views

Implicit equation that can not be parameterized. [on hold]

Is there an example of an implicit equation that can not be parameterized ?
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0answers
20 views

Finding regular values of $f(A)=A^2$

So I am trying to solve a few problems in differential geometry I found on the U of C website, and got stuck on this one: What are the regular values of the map $f: SO(3) \rightarrow SO(3)$ given by ...
3
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1answer
50 views

Clarifying notation used in Lie groups

Suppose that $G$ is a Lie group and $g \in G$ is a generic element. What does the notation "$dg$" refer to? Is it the differential of the function $G \to G$ given by left (or right) multiplication by ...
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1answer
20 views

Smoothness of the inversion map is redundant in the definition of Lie groups

The question I want to ask is different from this one. Let $M$ be a smooth manifold which admits a group structure such that the multiplication map $m:G\times G\to G$ defined as $m(g, h)=gh$ for ...
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0answers
26 views

Given second fundamental form what is the geometric /topological invariant?

The Gauss Bonnet integrates the first and second forms into an elegant structure. But before that... If first fundamental form alone is given, a series of mutually bendables with isometric /intrinsic ...
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22 views

Given a first fundamental form, showing a particular second form cannot exist

If I have a first fundamental form $ \mathrm{d}u^2+\cos^2 u \mathrm{d}v^2$, I am trying to show that the second fundamental form cannot equal $f(u,v)\mathrm{d}v^2$ for a smooth function $f(u,v)$. I ...
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1answer
51 views

Ellipse's farthest point to another point

I am trying to find the farthest and closest points of a ellipse without using any brute force type of coding. The processing power is limited so it should be as pinpoint as possible. I have tried a ...
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0answers
18 views

Simplification of vector cross product

If I have $-\mathbf{N}_u = a\mathbf{x_u} + b\mathbf{x_v}$ $-\mathbf{N}_v = c\mathbf{x_u} + d\mathbf{x_v}$ Then, why is $\mathbf{x_u} \times \mathbf{N_v} + \mathbf{N_u} \times \mathbf{x_v} = ...