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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Lie bracket of a pair of vectors at a point - is this a mistake in the book?

I've been reading Nakahara's "Geometry, Topology and Physics" and found something quite strange on the section 10.3.3 which discusses the geometrical meaning of the curvature of a connection. It is ...
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18 views

Show Lie bracket left invariant

I want to prove from the definition that the Lie bracket $[X,Y]$ of two left-invariant vector fields $X,Y: G \rightarrow TG$ where $G$ is a Lie group is again left-invariant. Left-invariance ...
4
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1answer
40 views

Can a volume form on a submanifold be extended to a parallel form in a neighbourhood?

Let $(M^{n+1},g)$ be a Riemannian manifold and let $\Sigma^n \hookrightarrow M$ be a smooth, closed, embedded submanifold. Let $\Omega$ be the volume form of $\Sigma$. It is well-known that a volume ...
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1answer
22 views

Derivative of a group action

Let $\phi : G \times M \rightarrow M$ be a group action on a smooth manifold $M$ and Lie group $G$. Then we define $$f(t):=\phi(g(t),d(t)).$$ where $g: I \rightarrow G$ and $d: I \rightarrow M$ are ...
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22 views

A manifold is a covering space over its quotient by a group action

Let $M \times G\to M$ be a properly discontinuous, free action of group $G$ on a manifold $M$. The quotient topology of the orbit space is Hausdorff. Suppose $p\in M$. How can we choose an open ...
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2answers
43 views

Construct a diffeomorphism $\phi:M\to M$ such that $\phi(p)=q$ and also $d\phi(X_p)=Y_q$

PROBLEM: Construct a diffeomorphism $\phi:M\to M$ such that $\phi(p)=q$ and also $d\phi(X_p)=Y_q$, where $M$ is a connected smooth manifold and $p,q \in M$ , $X_p \in T_pM$ and $Y_q \in T_qM$ I know ...
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1answer
42 views

Surface area of a slightly deformed sphere

Consider the unit sphere, which can either be described by $x^2+y^2+z^2=1$ or by the equation $r(\theta,\phi)=1$, where $(r,\theta,\phi)$ are spherical polar coordinates. I define a deformed sphere ...
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11 views

Divergence and formal adjoint operators: are they bounded/continuous?

Let $(M,g)$ be a smooth Riemannian manifold. The divergence operator is the map \begin{align*} \delta_g:\Gamma^k(S^2M)&\rightarrow\Gamma^{k-1}(T^*M)\\ ...
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2answers
25 views

On singular points of parallels

Say $\gamma$ is a unit speed curve and its parallel is given by $$ p (t) = \gamma (t) + d n(t)$$ where $n$ is the unit normal vector and $d$ is some scalar. I read that The parallels of a ...
2
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2answers
30 views

On reparametrisation of curves (sorry for trivial question but I'm confused)

I'm confused about speed and reparametrisations of curves. To illustrate my confusion please let me elaborate using the simplest example I could think of: Let $\gamma : [0, 2 \pi ) \to \mathbb R^2$ ...
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1answer
14 views

Parallels of a parameterised curve if not unit speed

I just read that if $\gamma$ is a curve given in unit speed parametrisation then the parametrisation of a parallel curve is given by $$ p(s) = \gamma (s) + d n(s)$$ where $n$ is the unit normal to ...
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0answers
36 views

Abstract definition of a differential operator

In Kolar, Michor, and Slovak, a differential operator is said to be a rule transforming sections of a fibred manifold $Y \to M$ into sections of another fibred manifold $Y' \to M'$. Is this is a ...
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0answers
41 views

What's book that I should read? [on hold]

When I read the chapter 4 of "Three Manifolds with Positive Ricci Curvature," I got stuck. I don't know what Fourier transform variable is, what derivative of second order nonlinear partial ...
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0answers
39 views

Map of smooth manifolds

Let $M$ and $N$ be smooth, connected $n$-dimensional manifolds. Let $M$ be compact and non-empty. Show that every embedding $f: M \to N$ is a diffeomorphism. So because $f$ is a embedding we have ...
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0answers
28 views

Derivative group action

Let $\phi : G \times M \rightarrow M$ be a group action on a smooth manifold $M$ and Lie group $G$. Then we define $$f(t):=\phi(g(t),d(t)).$$ Now I'd say: $$f'(t) = D\phi(g(t),d(t))(g'(t),d'(t)).$$ ...
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2answers
83 views

On a scale of 1 to 10 how far is this manifold from being a normal bundle?

(DIS)-CLAIMER: All the manifolds considered in this post are completely humble and have no additional structure beyond the smooth structure. Let $Y$ be a submanifold of $M$ and let $(-)^0$ denote the ...
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1answer
55 views

Advanced calculus: Solving quaternion differential equations

I have a system of two differential equations $$\frac{\partial X(t)}{\partial t}=a_1 A X(t)+a_2X(t) B+a_3 C Y(t)+a_4Y(t) D+a_5$$ $$\frac{\partial Y(t)}{\partial t}=b_1 E X(t)+b_2X(t) F+b_3 G ...
1
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1answer
41 views

Integral curves on immersed submanifold

An exercise of the book "Introduction to smooth manifolds - John M. Lee" asks to prove that if $S$ is a closed embedded submanifold of a manifold $M$, and $X$ is a vector field on $M$ tangent to $S$, ...
3
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0answers
22 views

The lie bracket as a limit

Let $M$ be a manifold and $f_t$ be the monoparametric group of local transformations generated by a vector field $X$. Suppose $w$ is a $k$-form and $x_1,\dots,x_k$ are vector fields. How can we see ...
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0answers
30 views

a question about Fenchel's theorem(differential geometry)

I am an undergraduate student studying differential geometry right now. I am just finishing reading how to prove Fenchel's theorem:The total curvature of a smooth closed curve in 3-dimensional space ...
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1answer
35 views

second fundamental form and connection forms

I am reading this paper that has the following: Suppose $M$ is an (n-1) dimensional closed hyper surface immersed in $\mathbb{R}^{n}$. Let $e_1, \cdots, e_n$ be orthonormal frame in $\mathbb{R}^n$ ...
5
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53 views

how much differential structure can we put on countable manifolds?

The motivation for this question is that I would like to formulate Lagrangian mechanics in a purely discrete setting (see also my older question at physics.se). Unfortunately several key pieces of ...
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0answers
27 views

Let $p=(5,0,-4)$ and $v \in T_{(5,0,-4)}M$. Compute $(F^{*}\omega)_p(v)$.

Let me show my work before presenting the problem itself. Let $M=\{(x,y,z) \in \mathbb{R}^3 : x+y=5, x+z=cos^2y\}$. We can easily see that $M$ is a submanifold of $\mathbb{R}^3$ of dimension $1$. ...
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2answers
42 views

What am I doing wrong? Derivative of pedal

Let $\gamma$ be a unit speed curve $\gamma : I \to \mathbb R^2$. The pedal is given by $P (s) = (\gamma (s) \cdot N(s)) N(s)$. I tried to calculate the derivative as follows: $$ P' = (\gamma N)' N ...
4
votes
1answer
38 views

Parametrizations and coordinates in differential geometry - what's the difference?

From what I've read one can introduce the notion of a tangent vector to a point on a manifold in terms of an equivalence class of curves passing through that point (the equivalence relation being that ...
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35 views

Natural derivative of Vector Fields on manifolds

I'm learning about connections and my book says that there is no natural derivative for a vector field on a manifold. Wouldn't it be possible to cook up a connection by just letting $\nabla_{v_p}X = ...
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1answer
66 views

Connection between harmonic functions, Bochner Laplacian and Ricci curvature

I stumbled upon the following claim in a paper: "We write the (Bochner) Laplacian in suffix notation: $\Delta_B = \nabla ^k \nabla_k$". after this statement, the following is written: ($M$ is a ...
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0answers
18 views

Normal Vectors to Action of Orthogonal Group

Let $X\in\mathbb{R}^{n\times r}$ be a fixed matrix with orthogonal columns, and let $U\in\mathbb{R}^{n\times r}$ be given. Because the group of orthogonal $r\times r$ matrices, $O(r)$, is a compact ...
4
votes
1answer
29 views

Making a bijection into a diffeomorphism

Given a set $M$, one that can be made into a smooth manifold, and a bijection $f:M\to M$, does there exist a differentiable structure on $M$ such that $f$ is a diffeomorphism? In case it's not always ...
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3answers
201 views

Is a ball noncompact?

A compact manifold usually refers to "a manifold without a boundary", for example the usual 2-sphere $S^2$. What about a manifold with a boundary? Intuitively, I think such an example, e.g. a ball ...
2
votes
1answer
39 views

Unit sphere and Ricci curvature

Why is it that on the unit sphere the Ricci curvature Ric = g (where g is the metric defined on the unit sphere) ?
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1answer
30 views

Determine the lines of curvature of $z=xy$

I have to find the lines of curvature of $z=xy$ I calculate Weingarten Matrix as described below $p_u = (1, 0, v),p_v=(0,1,u),\nu =\frac{1}{\sqrt{1+u^2+v^2}}(-v,-u,1)$ so, $E=1+v^2,F=uv,G=1+u^2$ ...
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2answers
196 views

Nonexistence of local isometry between equidimensional Riemannian manifolds

Recall that all inner product spaces of the same dimension are isometric. For example, if $(M,\mathrm{g})$ and $(N,\mathrm{h})$ are Riemannian manifolds of the same dimension, then ...
2
votes
1answer
33 views

Compute in the chosen charts of $M$ and $S^1$ the expression of $DF_{(5,0,-4)}$

Let me show my work before presenting the problem itself. Let $M=\{(x,y,z) \in \mathbb{R}^3 : x+y=5 x+z=cos^2y\}$. We can easily see that $M$ is a submanifold of $\mathbb{R}^3$ of dimension $1$. We ...
2
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0answers
27 views

Action of $H^1$ on spin structures

Since the set of spin structures on a principal $SO(n)$-bundle on a manifold $X$ is in one to one correspondence with $H^1(X,Z_2)$, this group admits an action on spin structures. I wanted to know if ...
0
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1answer
28 views

Lagrange multipliers and critical points (differential form description).

On $M \times V^*$, where $M$ is a differentiable manifold (not necessarily equipped with a metric) and $V^*$ is dual to a vector space $V$, one can define a Lagrange function $F = f +v^*h(x)$ using ...
4
votes
1answer
31 views

What is the difference between Nakano Postivity and Griffiths Positivity of Hermitian vector bundles?

I am currently reading "Complex Differential Geometry" by FY Zheng on the curvature of Hermitian vector bundles. In section 7.5, he described a Hermitian vector bundle $(E,h)$ over a complex manifold ...
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1answer
24 views

When do we call the boundary $\partial\Omega$ of a bounded domain $\Omega\subseteq\mathbb{R}^n$ smooth?

When do we call the boundary $\partial\Omega$ of a bounded domain $\Omega\subseteq\mathbb{R}^n$ smooth? I can't find a formal definition. I know, that we say, that $\partial\Omega$ has a ...
4
votes
1answer
25 views

principal curvature of the flat torus

I am looking at the Hopf-fibration and I am looking at the preimage of the equator in $\mathbb{S}^2$. I think that I have proved that this is just the flat torus and now I want to calculate the ...
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vote
1answer
44 views

Injectivity of the Differential of Smooth Map

I am trying to answer the following question: Let $M = \{(x,y)\in \mathbf{R}^2 : x^2 + y^2 < 1\}$. Define a smooth or $C^\infty$ function by $f\colon M \rightarrow \mathbf{R}^2$ as ...
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0answers
32 views

Kinds of isometries preserving the curvature tensor

We talked about isometries /local isometries and linear isometries in the lecture, but unfortunately we did not say when the isometry preserves the curvature tensor or sectional curvature. First I am ...
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1answer
28 views

Tensor Product of Vectors

let $S,T$ be respectively $k-, n- $ tensors; $k,n>0$. Then we define the tensor product $$ T \otimes S(x_1,x_2,....,x_{k+n}):=T(x_1,...,x_k)S(x_{k+1},...,x_{k+n}) $$ (their product as Real numbers, ...
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1answer
40 views

Tangent space as derivations exercise

Thinking of the tangent space to a manifold as derivations is a concept which just kind of eludes me. I am comfortable thinking about tangent vectors as equivalence classes of curves and with the ...
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1answer
25 views

Manifold with boundary given as the pre-image of a subset of $\mathbb{H}^n$

Let $f \colon \mathbb{R}^{n+k} \to \mathbb{R}^n$ be a function of class $C^r$ for $r>1$. If $M = f^{-1}(0)$ and $0$ is a regular value of $f$, then we know (using implicity functions theorem) that ...
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2answers
115 views

What's my mistake in the calculation?

Summation convention holds. If $\frac{\partial}{\partial t}g_{ij}=\frac{2}{n}rg_{ij}-2R_{ij}$, then ,I compute: $$ \frac{1}{2}g^{ij}\frac{\partial}{\partial ...
3
votes
1answer
26 views

Connection form on a frame bundle

Let $(E(M),\pi,M)$ be the frame bundle over a manifold $M$ of rank $n$. Consider a covering of $M$ by open neighborhoods $U_{\alpha}$. Let $s_{i}$ and $t_{j}$ where $i,j\in {1,...,n}$ be frames of $M$ ...
3
votes
2answers
77 views

Gaussian Curvature of $x^4+y^4+z^4=1$

Let $S=\{(x,y,z)\in \mathbf R^3 | x^4+y^4+z^4=1 \}$ . To compute the Gaussian curvature $k$ of $S$, I tried an elementary method to find $dN_p$. Let $\alpha (t) = (x(t),y(t),z(t))$ be an parametried ...
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1answer
21 views

$M\subset\mathbb{R}^n$ is a open subset, $p\in M$ is arbitrary. Find $T_pM$ and $N_pM$

PROBLEM: $M\subset\mathbb{R}^n$ is a open subset, $p\in M$ is arbitrary. Find $T_pM$ and $N_pM$. I know how to determine $T_pM$ and $N_pM$ for explicit examples, but I dont know how to handle this ...
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0answers
21 views

Vector field from group action

Let $\Phi: G \times \mathbb{R}^4 \rightarrow \mathbb{R}^4$ be a group action where $G = \mathbb{R}/(2 \pi \mathbb{Z}).$ Then $$\Phi(\theta,(x_1,x_2,p_1,p_2)) = ( R(\theta) (x_1,x_2)^T, R(\theta) ...
3
votes
2answers
80 views

Boundedness of the norm of the Riemann curvature tensor

Let $(M,g)$ be a Riemannian manifold and let $R(X,Y)Z$ be its $(3,1)$ Riemann curvature tensor given by $$R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z$$ Let the input vectors $X,Y,Z$ ...