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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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
10 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
21 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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1answer
11 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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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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1answer
14 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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2answers
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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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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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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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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45 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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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
7 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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4 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 ...
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1answer
36 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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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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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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40 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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1answer
42 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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1answer
20 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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28 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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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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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 ...
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1answer
45 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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1answer
36 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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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
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. ...
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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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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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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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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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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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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 ...
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37 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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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
19 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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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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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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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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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} = ...
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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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A space curve with non-vanishing curvature is planar iff its torsion is 0

Intuitively this is simple and to prove the backwards direction: $\tau = 0 \Rightarrow \mathbf{b'}=0 \Rightarrow \mathbf{b} $ is constant. Then letting $\gamma$ be the parametrisation of the curve ...
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1answer
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A lie group $G$ is compact iff $G/H$ is.

Suppose that $G$ is compact Lie group and $H$ is closed subgroup. Than $G/H$ is compact since canonical projecton is continuous. Is the inverse true, namely can one prove that: G is compact iff ...
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1answer
38 views

Kato's inequality

Let u be a smooth function defined in a Riemannian manifold $(M,g)$. The well known Kato's inequality states $$|∇|∇u||^2≤|∇^2u|^2$$ where $∇^2$ represents the Hessian operator of $M$. I would like ask ...
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curvature of an arc in S3, in stereographic projection

$r(t)$ is a unit 4-vector. The derivatives of $r$ are known and well-behaved. I'm interested in images of $r$ in stereographic projection – but (for purposes of this question) I don't yet know where ...
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2answers
68 views

Circular cylinder $S=\{ (x,y,z) : x^2+y^2=1 \}$ can be covered with a single surface patch.

I somewhere found that we can take $U$ an annulus instead of a disc where $U=\{ (u,v): 0 < u^2+v^2 < π \}$. Can anyone please explain me that how a cylinder can be covered with a single surface ...
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a algebraic concept in reimannian geometry?

let $(M,g)$ is a $C^{\infty}$ manifold with product form of $M=R \times \Theta $ the for each element $\varepsilon (r,\theta )\in M$, the tangent space $T_{\varepsilon}M$ is natural isomorphic to ...
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35 views

Is there any large diffeomorphisms of $S^{n}\times S^1 $like Torus?

We know that a Torus is mapped onto itself in a special discontinuous transformation given by $PSL(2,\mathbb{Z})$. Thinking of torus as $S^{1}\times S^{1}$ and thus as a lattice, we can easily show ...
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Equivalent conditions for $\mathfrak{F}$ to be a differential ideal

Heres the question: Let $\mathfrak{F}$ be an ideal of forms on a manifold $M$ locally generated by $r$ independent $1$-forms. Say $\mathfrak{F}$ is generated by $\omega_1,\ldots, \omega_r$ on $U$. ...
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Maps between tangent space of product manifold and sum of tangent spaces

I am trying to prove that $$T_{(p,q)}(M\times N)\cong T_pM\oplus T_qN$$ We define: $$\Phi:T_{(p,q)}(M\times N)\to T_pM\oplus T_qN:v\mapsto(d_{(p,q)}\pi_M v,d_{(p,q)}\pi_N v)$$ and $$\Psi:T_pM\oplus ...