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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Vector valued 2-forms wich satisfy Jacobi Identity

Motivated by this MO question we ask the following two questions: What is an example of a compact manifold $M$ which does not admit any (admit at least one) smooth (1,2) tensor $\omega$ which ...
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A Sub-Laplacian equal to $\displaystyle\sum_{k=1}^n \partial_{x_k}^2$

Let $\mathbb G=(\mathbb R^n, \star)$ be Homogeneous Lie group (or a stratified Lie group) and let $L$ be a subLaplacian on $\mathbb G,$ defined by $$ L_{\mathbb G}=-\sum_{1}^{n}X_{i}^{2} $$ ...
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Trigonometric parametrization of a genus g surface?

It is possible to find functions $\phi, \psi \in \mathbb{R}[sin(x), sin(y), cos(x), cos(y)]$, so that $S^2 = \phi( [0,1]^2)$ and $\psi( [0,1]^2)$ is a torus. Is it possible find, for any genus g, ...
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480 views

Geodesics on torus

Describe the geodesics on Torus $$\sigma (u,v)= ((a+b \cos u)\cos v, (a+b\cos u)\sin v, b\sin u)$$ First fundamental form for torus is $$b^2 du^2 +(a+b \cos u)^2dv^2$$ Consider unit-speed ...
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Differential topology versus differential geometry

I have just finished my undergraduate studies. During last two semesters I've taken two subjects dealing with manifolds: Analysis on manifolds, containing: definition of manifold, tangent space (as ...
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56 views

commuting property of connections and bundle homomorphisms

I have the following situation: $E,F$ are (smooth) vector bundles over a smooth manifold $M$. Assume we are given connections $\nabla^E,\nabla^F$ on $E,F$ and a homomorphism of $M$-bundles ...
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Definition of covering (deck) transformation for smooth manifolds: Are they diffeomorphisms?

In John Lee's book Riemannian Manifolds, a covering transformation (or deck transformation) of a smooth covering map $\pi:\tilde{M}\to M$ (of connected smooth manifolds) is defined to be a smooth map ...
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Second variation formula and Jacobi fields

Let $\bar{\alpha}:U\rightarrow \Omega$ be a two-parameter variation of a geodesic $\gamma$. For $i=1,2$ we define $$W_{i}=\frac{\partial \bar{\alpha}}{\partial u_{i}} \in T_{\gamma}\Omega$$ the ...
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Jacobi field strange condition.

I am currently reading a textbook (Kuehnel) saying that if $V,W \in T_pM$ are such that $\langle V,W \rangle =0$ and $\|V\|=\|W\|=1,$ then $Y(t):=D \exp(tV)(tW)$ is a Jacobi field. The thing is, I ...
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A theorem of Cartan

My question is regarding when the polarmap defiened from an isometry $L:T_pM\rightarrow T_{\bar{p}}N$ extends to an isometry of the normal neighborhhods on which it is a diffeomorphism. I have come ...
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54 views

Homeomorphism between $S^2$ and $CP^1$ via uniqueness of quotient

I am trying to show that $S^2$ and $\mathbb{C}P^1$ are homeomorphic making use of the following result - see e.g. Jack Lee Introduction to topological manifolds. Let $Y \xrightarrow{\pi_1} X_1 $, $Y ...
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Curvature exercise, need hint.

Let $\alpha: \mathbb{R} \to \mathbb{R}^3$ be a parametrization of a smooth curve in space with $\|\alpha'(s)\| = 1$ constant. Suppose that for all $s$ within a sufficiently small neighborhood of fixed ...
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41 views

Is there a unique preferred connection on a general manifold?

I was wondering whether, for a given finite-dimensional manifold, the connection $\nabla$ exists and is uniquely defined? Afais for Riemannian manifolds, there exists always exactly one Levi-Civita ...
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Exercise about Hodge Star Operator on $\Lambda^{p,q}$

In real case, $$ \ast (e_1\cdots e_k)=e_{k+1}\cdots e_n$$ on $\mathbb{R}^n$, where $e_i$ is $1$-form and $e_1\cdots e_n$ is volume on $\mathbb{R}^n$ We will extend to complex case. Define $$ dz_k:= ...
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Map between $SO(n)$ is homotopic to the identity?

I'm given an exercise, in a differential geometry class, where I need to detemine wether or not the smooth map between manifolds: \begin{align} f \colon\ &SO(n) \rightarrow SO(n)\\ & A \mapsto ...
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23 views

Required minimum number of points on boundary of minimal surface

What is the minimum number of points required to uniquely determine a minimal surface in 3-Space? Four? Five? If six or more points on boundary are given it gives rise to over-determination.. right?
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Question on application of the Morse lemma

Consider the following example of an application of the Morse lemma (to plane function): Consider two unit speed pieces of curve (perhaps part of the same closed curve) parametrized by ...
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a differential geometry question, is $k_\alpha(s) \ge |k_\beta(\alpha(s))|{d\sigma\over ds}$ true or wrong?

let $ \alpha(s)$(s is the arc-length of $\alpha$) be a closed curve in three dimensional space, let $\beta$ be orthogonal projection of $\alpha(s)$ in xy plane. if so, we have ...
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52 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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31 views

Definition of submanifolds by regular values

Let $f: M \rightarrow N$ and $q \in N$ be a regular value, then $f^{-1}(q)$ is a submanifold of $M$. Now assume that $q \in N$ is not a regular value, but you pick $K:=f^{-1}(q) \cap \{p \in M; ...
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Symbol $\Gamma$ when talking about vector fields.

I noticed several times online that people tend to use the symbol $\Gamma(M,TM)$ when talking about the space of smooth vector fields on smooth manifolds. I find this totally confusing, as in ...
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51 views

Upper bound on the distance of orthogonal matrices

Dear math stackexchange users, I have a question on orthogonal matrices: suppose I have a matrix $X\in\mathbb{R}^{n\times n}$ and I consider the orbit of the orthogonal group $O(n)$ acting from the ...
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Research areas lying at the confluence of Analysis and Geometry [on hold]

I wanted to get expert opinion on what are the areas of active research lying at the confluence of Analysis and Geometry. Two areas that I have heard about are : (1)Geometric Analysis and ...
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Scaling of minimal surfaces

After scaling and suitable Euclidean motions every rigid minimal patch can be placed on a unit catenoid of revolution $ x^2 + y^2 = c^2 \cosh^2 (z/c), c=1.$ with full area contact. Is the statement ...
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86 views

Statement about the isometries of a product manifold

I'm studying Minkowski spacetime $\Bbb{M}$, and I would like to make the following statement about its symmetry transformations. Since $\Bbb{M}$ is the product manifold of time and space, it inherits ...
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2answers
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Wrong pushforward of vector field definition on wikipedia

Wikipedia claims that for $$\mathrm d \varphi_x:T_xM\to T_{\varphi(x)}N\,$$ we have for $X \in T_pM$ and $f \in C^{\infty}(N,\mathbb{R})$ $$\mathrm d\varphi_x(X)(f) = X(f \circ \varphi)$$ whereas I ...
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41 views

What is the formal name for the conformal laplacian?

\begin{align} L=R-4\dfrac{n-1}{n-2}\nabla^k\nabla_k \end{align} What is the formal name for $L$? I have seen it referred to as the conformal laplacian, however I thought I once read $L$ with a formal ...
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16 views

Significance of sasaki manifold, kenmotsu manifold, einstein manifold and different curvature tensors

I have studied different sasaki manifold, kenmotsu manifold and einstein manifold and also different curvature tensors during my masters and also worked on the above domains. However working on ...
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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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equivalent definitions of tensorfields

I have been confused about something for quite some time now and I would really much appreciate to get a clear explanation of the following. There are two equivalent definitions of how to define ...
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45 views

Intuition behind eigenfunctions of the Laplacian operator

I'm reading about the notion of spectral dimension which is a measure of how particles diffuse in some space at different scales. An important aspect of spectral dimension is the ...
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The equation of hypersufaces in $R^{4}$ [on hold]

Can this be an equation of hypersurface in $R^{4}$: $x_{1}+x_{2}=c$, $c$=const.
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47 views

Research in Non Commutative geometry [on hold]

I am currently doing my Masters' in Mathematics and I wish to pursue Ph.D. . I have taken courses in Differential Geometry of Manifolds and C* Algebras, and some introduction to Riemannian geometry. I ...
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1answer
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$\bigwedge^k T^*M$ is a $\binom{n}{m}$-dimensional Subbundle of $\bigotimes^k T^*M$.

I am trying to prove the following: Let $M$ be a smooth manifold. Then $\bigwedge^k T^*M$ is a smooth subbundle of dimension $\binom{n}{k}$ of $\bigotimes^kT^*M$. To do this, I think the ...
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2answers
50 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 $S$, ...
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127 views

On the variation of a Kähler metric on a surface by pullback of the complex structure

Let $\Sigma$ be a compact, connected, oriented surface, and let $\rho\in\Omega^2(\Sigma)$ be a fixed volume form. Then any (almost) complex structure $J\in\Omega^0(M;\operatorname{End}TM)$ compatible ...
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Complement of a foliation

I have an $n$-manifold $M$ which is foliated by leaves $F_\alpha$ of dimension $p$ and a path $\gamma:[0,1]\to M$. You can take without problems $\gamma$ to be injective. Is the following statement ...
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1answer
26 views

Condition equivalent to a moduli space being a manifold

Let $M$ be an $n$-manifold, and assume that it is foliated by a regular $p$-foliation. I know the following implication to be true: If for every point $m\in M$ there exists a submanifold $m\in ...
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1answer
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Choosing a vector normal to a jordan curve that points “inside”

Let $\gamma=\partial K_1(0,0)$ be the circle with radius $r=1$ and origin $(0,0)$ in $\mathbb R^2$. Then for any $t_0$ we have $\gamma'(t_0)\neq \begin{pmatrix} 0 \\ 0\end{pmatrix}$. Let ...
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1answer
29 views

Unit radial vector field

Lee's book defines the unit radial vector field in normal coordinates as $$ \partial_r:= \frac{x^i}{r(x)} \partial_i$$ and $r(x):=\sqrt{\sum_i (x^i)^2}$ Now this is a unit vector field iff ...
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3answers
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The Euclidean Metric on $\mathbf R^3$ Induces an Index-Lowering Isomorphism $b:\mathfrak X(\mathbf R^3)\to \Omega^1(\mathbf R^3)$.

In Lee's Introduction to Smooth Manifolds, Second Edition, the line just before Equation 14.25 reads The Euclidean metric on $\mathbf R^3$ induces an index-lowering isomorphism $b:\mathfrak ...
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1answer
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Is a principal bundle automorphism locally given by a left action?

Let $G\hookrightarrow P \xrightarrow{\pi} M$ be a principal bundle, denote by $\cdot$ the right action of $G$ on $P$. Let $f:P\rightarrow P$ a bundle automorphism (i.e. $f$ is a diffeo, $f(p \cdot g) ...
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Lie bracket and flows on manifold

Suppose that $X$ and $Y$ are smooth vector fields with flows $\phi^X$ and $\phi^Y$ starting at some $p \in M$ ($M$ is a smooth manifold). Suppose we flow with $X$ for some time $\sqrt{t}$ and then ...
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Describe the local flux of this vector field $X$ on $S^2$ given by $X(v)=w_0- \langle v, w_0 \rangle v.$

Let be $w_0 \in \mathbb{R}^3$ and $X: S^2 \rightarrow TS^2$ the vector field on $S^2$defined by: $$X(v)=w_0- \langle v, w_0 \rangle v.$$ ($\langle.,. \rangle$ is the standard dot product) How can I ...
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1answer
23 views

Trouble understanding holonomic dual basis

I saw a post earlier and it made me a bit confused, so I looked into it and I have some questions about a paper I saw. It said since a tangent vector can be defined $u=d/dλ$ where λ is the parameter ...
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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
27 views

Four Vertex Theorem

In Do Carmo's "Differential Geometry of Curves and Surfaces" he has a proof of the four vertex theorem that I am having trouble getting my head around. In it, he starts by assuming a closed, simple, ...
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63 views

Is this operation meaningful or it is 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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71 views

Parallel transport of a vector in hyperbolic space, specifically in $\mathbb{H}$

Let us consider Poincaré's upper plane which is defined as $\mathbb{H} = \{ (x,y) | y>0\}$. This space has a Riemannian metric $g = \text{diag}(1/y^2, 1/y^2)$. Now let us consider a differential ...
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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 ...