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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Understanding tensors

Locally in a chart, a tensor field looks like $$T= T^{i_1,...i_n}_{j_1,...,j_m} dx^{j_1} \otimes...\otimes dx^{j_m} \otimes \partial_{i_1} \otimes ... \otimes \partial_{i_n},$$ where ...
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Is the dimension of a smooth manifold an invariant of the underlying set in it?

Let $M$ be a smooth manifold, $S$ a set and $f:M\to S$ a bijection (assuming of course, that such a function does exist). It's an easy exercise to show that $S$ can be given a differentiable ...
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24 views

Algebraic vs Geometric Picard groups

Given a smooth manifold $P$. I found two definition of a Picard group. The firs one is algebraic, $Pic(C^{\infty}(P)),$ defined as the set of self-equivalence bimodules (Morita equivalence) with ...
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How to reduce cubics in the plane to a canonical form?

I watched a video from Wildberger in the Differential Geometry series ( first, or third lecture, I don't remember ) where he says the following. The general format of a cubic curve is $$a x^3 + b ...
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1answer
46 views

Arc length function of a helix/spiral is convex?

Given the arc-length of a parametric curve, $\int_a^b\|\gamma'(t)\|$ if the parametric curve was non-convex, can the arc length be a convex function?If the parametric curve was convex, will the arc ...
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1answer
35 views

Closed form defines locally a function?

In my particular example, I have a volume form on a one-dimensional manifold, so i.e. this volume form $dx$ is closed. Now, I was wondering, does the integral function $f(y):=\int_{x_0}^{y} dx$ define ...
3
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1answer
43 views

Covariant and partial derivative commute?

I know that we have for a function $\Gamma: (-\varepsilon,\varepsilon)^2 \rightarrow M$ we have (at least I think I know that this is true) $$\nabla_{\frac{\partial \Gamma}{\partial s}} ...
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69 views

What is the Exterior Derivative Trying to Do?

$\newcommand{\R}{\mathbf R}$ Consider a smooth function $f:\R^n\to\R$ and let $Df:\R^n\to \R^{n*}$ be the map which takes a point $\mathbf a\in R^n$ to the linear map $Df_{\mathbf a}:\R^n\to \R$. ...
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31 views

Jacobi fields and variations

I want to show that every Jacobi field is a variation of geodesics, i.e. let $Y: I \rightarrow TM$ be a Jacobi field along a geodesic $\gamma$, then I want to show that $Y$ can be written as a ...
2
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2answers
28 views

Local existence of parallel vector field

Let $M$ be a Riemannian manifold, $p\in M$ be a point in the manifold, and $\xi\in T_p M$ be a vector in its tangent space. I am wondering whether, for some small neighborhood $U\subset M$, it is ...
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18 views

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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Vector valued 2-forms wich satisfy Jacobi Identity

Motivated by this MO question we ask the following two questions: 1)What is an example of a compact manifold $M$ which does not admit any smooth (1,2) tensor $\omega$ which restriction to each ...
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15 views

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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1answer
95 views

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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1answer
41 views

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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24 views

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 ...
3
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2answers
52 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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22 views

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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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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2answers
34 views

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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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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2answers
79 views

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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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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2answers
52 views

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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25 views

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 ...
3
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1answer
34 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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1answer
56 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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0answers
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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22 views

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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27 views

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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1answer
48 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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51 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 ...
6
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1answer
174 views

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 ...
2
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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 ...
0
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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 ...
3
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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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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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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 ...
0
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1answer
29 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, ...
5
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1answer
54 views

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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2answers
75 views

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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1answer
22 views

Diffeomorphism between covering spaces

Let $\pi_1: M \rightarrow M_1$ and $\pi_2: N \rightarrow M_2$ be two smooth covering maps. Now $\phi: M \rightarrow N$ is a smooth diffeomorphism. Does this induce a smooth diffeomorphism $f: M_1 ...
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Is there a relation between vectors on these two spaces?

I've been reading lately one paper on Physics, which basically presents one gauge theory approach to the problem of swimming at low Reynolds number. I've been trying lately to rewrite some of the ...
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1answer
35 views

Calculating euler characteristic and geodesic curvature

We have the usual formula for the euler characteristic in differential geometry $$\chi = \frac{1}{4\pi}\int_{M}d^{2}\sigma g^{1/2}R + \frac{1}{2\pi}\int_{\partial M}ds k$$ where we define the ...
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Problem of Arnold's book and covering spaces

I am currently reading Arnold's book "Mathematical Methods of classical mechanics" on page 278 and I don't see through his arguments there at a point. Especially, I am talking about the part that ...
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56 views

Properties of $\mathbb{C}P^n$

I'm currently working on a somewhat deformed version of $\mathbb{C}P^2$ and want to check some properties from a geometrical and/or topological point of view. Of course, $\mathbb{C}P^2$ is Kähler ...
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1answer
39 views

Application of inverse function theorem?

I am not completely sure if this a direct consequence of the inverse function theorem. Assume that we have a function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ that we can write in terms of ...
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1answer
42 views

$\bigwedge^k T^*M$ 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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1answer
18 views

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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Preimage of a small normal deformation under an embedding again a normal defomation?

Let $j\colon M\rightarrow W$ be a smooth embedding of smooth manifolds and assume $M$ and $N$ have Riemannian metrics, but $j$ is not necessarily an isometric embedding. Let $N\subseteq W$ be a ...