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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Relationship between differential cohesion and synthetic differential geometry

I am wondering what is the relationship between differential cohesion and synthetic differential geometry? Is the latter a vertical categorification of the theory of synthetic differential geometry? ...
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Multiplying two tensors of the Levi-Civita type [on hold]

How to multiply two epsilons with one another? We know ...
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1answer
79 views

Is parallel transport injective?

For a vector bundle $E\to X$ with a given connection $\nabla$. We say that a section $s$ of $E$ is parallel to a vector space $V$ if $\nabla_V s=0$. If $\gamma:[0,1]\to X$ is a smooth path, we say ...
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44 views

Are closed simple curves with that property necessarily circles?

This is a more interesting follow-up to the question Are closed simple curves with this property necessarily circles? Let $\gamma:[0,1]\to \mathbb R^2 $ be a closed simple $C^1$ convex curve and ...
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50 views

Understanding a step in Yi Fang's Lectures on Minimal Surfaces

In Yi Fang's Lectures on Minimal Surfaces, page $94$, there's a step that I didn't understand, and that perhaps is wrong. I'll estabilish some notation first. We have that $X$ is a minimal surface, ...
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18 views

Envelope of a family of lines. When does it exist?

Given a family of lines $(\gamma_t): a(t)x+b(t)y+c(t)=0,\ t\in I$ (interval in $\mathbb{R}$), where $ a,b,c:I\to\mathbb{R},\ a^2(t)+b^2(t)\neq 0,\ \forall t\in I$ , what conditions should we impose ...
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Lie groups and Lie algebras for matrices

Recently, I stumbled over a few things in very basic Lie group / Lie algebra theory concerning matrix groups. Basically, my question is: Is there a way to canonically understand all the Lie groups ...
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how to embed a square into $R^2$?

By Whitney embedding theorem you can embed a smooth 1-manifold in $\mathbb{R}^2$. Now if you give the unit square a smooth structure(for example by inducing the unit circle's smooth structure on it), ...
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14 views

How can we define regular curves implicitly?

Let $F:D\subseteq\mathbb{R}^2\to\mathbb{R}$, $D$ open and connected set, be a $C^1 (D)$ application. What are the minimum requirements for $F$ such that the solutions of the equation $F(x,y)=0$ are ...
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1answer
25 views

Special linear group as a submanifold of $M(n, \mathbb R)$

I have that $SL(n,\mathbb{R})$ is an embedded submanifold of dimension $n^2-1$ in $GL(n,\mathbb{R})$, and I know that $T_XGL(n,\mathbb{R})$ is isomorphic to $M(n,\mathbb{R})$ for all $X \in GL(n , ...
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determinant of general linear group

I know that for the general linear group, the coordinate derivatives of the determinant function $\det:GL(n,\mathbb{R})\to \mathbb{R}$ are \begin{equation*} \frac{\partial}{\partial X^i_j}\det X=(\det ...
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Example of locally symmetric spaces

A locally symmetric manifold is a manifold with parallel curvature tensor $\nabla R=0$. Can you give an example except spheres, projective spaces and hyperbolic spaces?
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57 views

Non-compactness of the resolvent

Consider a complete non-compact Riemannian manifold $M$ and the resolvent of the Laplacian $(-\Delta + \lambda I)^{-1}$. It is known that the resolvent is in general not a compact operator. I am ...
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1answer
40 views

Preimage of singular points of smooth map between manifolds

Given a smooth ($C^{\infty}$) map $\phi: V \rightarrow SU(n)$ where $V$ is a (finite dim, real) vector space (of potentially very large dimension) and $SU(n)$ is the special unitary Lie group, what ...
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1answer
22 views

Boundary preserving map

Let $K\subseteq\mathbb{R}^2$ be a compact set. Is it true that for a continuous map $p:K\to\mathbb{R}^2$ we have: $p(\partial K)=\partial p(K)$? Are there any generalizations? P.S. Note that ...
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17 views

Bianchi geodesic polar circles

Are circles with u = const in concentric geodesic polar coordinates, and Bianchi Circles: http://www.jstor.org/stable/1967629 one and the same? Earlier for this question here, the above ( century ...
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1answer
29 views

Definition of vector field along a curve

Let $γ : I→R^3$ be a regular parametrization of a curve C. If asked what a vector field on C is I would perhaps answer like this: 1) "a smooth function $v$ associating to any point $γ(t)$ of C an ...
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1answer
22 views

Finding surface of revolution isometric to helicoid

I'm trying to find a function $f(x)$ such that the two surfaces given below are isometric: $$f_1(x,y) = (ax \cos(y), ax \sin(y), y)$$ $$f_2(x,y) = (f(x)\cos(y), f(x)\sin(y), x)$$ Now I understand ...
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124 views

Is it possible to develop differential geometry without points?

I read about pointless topology and locale theory, and become curious about this topic. For example, there is the concept "differential manifold" corresponds to "topological manifold". As this, are ...
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1answer
32 views

Eigenfunctions of $-\Delta_{S^n}$

I read somewhere that the eigenvalues of the Laplacian $-\Delta_{S^n}$ on the sphere $S^n$ consist of $k^2 + (n - 1)k$, with the corresponding eigenspace $V_k$ consisting of homogeneous harmonic ...
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30 views

Transversality of graphs of functions

Consider the $C^1$ function $f: [0,1] \to \mathbb{R}$. I understand that a curve in the plane that intersects the graph of $f$ non-transversally would be tangent to it at a point of intersection. I ...
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38 views

Cohomology classes of the DeRham cohomology

May be $TM$ a tangent bundle of the manifold $M$ and $\wedge^n TM$ the set of all $n$-forms. The map $d: \bigwedge^n TM \rightarrow \bigwedge^{n+1}TM$ is called the exterior derivative and it holds ...
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26 views

What is an affline manifold? How do check if a space (?) is affline manifold. Is it related to vector space?

I have no idea about differential geometry or topology because I am engineer. I need the conditons to check the applicability of a theorem to particular case.From my search I could only find a wiki ...
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1answer
21 views

If $V$ irreducible an. variety $V=V_1\cup V_2$, then $V_1\cap V_2\subset V_{sing}$

Let $V$ be an irreducible analytic variety, and $V_1, V_2$ analytic subvarieties such that $$V=V_1\cup V_2.$$ In Griffiths-Harris book, it is mentioned that $V_1 \cap V_2$ is a subset of the ...
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When is a metric diagonalizable?

I'd like to understand a bit more the following problem. Suppose a potato shape (say a 3D volume bounded by a 2D surface), and define $n$ the normal vector to its surface, with components $n_i$. ...
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22 views

Gaussian curvature distribution: embeddable?

Given a smooth, rotationally symmetric Gaussian curvature profile, $G(r)$, how can we know if it is embeddable in $R^3$? $G(r)$ is defined on $r=[0,R)$ where $r$ is the geodesic length from a fixed ...
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28 views

Proving that $\phi$ is orthogonal to the harmonic forms given $\int\phi \;d\mathrm{vol}$.

I want to prove that given a connected closed (compact without boundary) oriented Riemmanian manifold $(M,g)$, the condition $$\int_M \varphi \;\mathrm{d}\mathrm{vol}_g=0$$ implies that $\int \varphi ...
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35 views

Unknown proof Lie Algebra

I have a calculation where I do not know what it actually shows. I think it tells me that for right invariant vector fields, the commutator is again right invariant. Maybe somebody here could help me ...
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14 views

Involute Frenet frame

So, a curve $C_1$ is called an involute of a given curve $C$ if tangents of $C$ are normal to $C_1.$ I'm wondering what can we say about whole Frenet frame for involute relatively to the curve?
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39 views

Integral of arc length $\int_0^{2\pi} \sqrt{(R+r \cos t)^2 +r^2} dt$

$$\int_0^{2\pi} \sqrt{(R+r \cos t)^2 +r^2}\, dt$$ Where $R > r$ and both are constants. This is all that I am looking to calculate, however I thought it would be nice to explain the context in ...
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How can we show that $S^0$ is a manifold?

Recall $S^n = \{ (x^0, ..., x^n) \in \mathbb{R}^{n+1}: {x^0}^2 + ... + {x^n}^2 = 1 \}$ $S^0$ is a very cute set on $\mathbb{R}$ consisting of points $\{-1, 1\}$. How can we show that it satisfies the ...
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1answer
22 views

Describe a parallel transport on sphere

Let $S^2\subset \mathbb{R}^3$ be the unit sphere, $c$ an arbitrary parallel of latitude on $S^2$ and $V_0$ a tangent vector to $S^2$ at a point of $c$. Describe geometrically the parallel transport ...
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1answer
47 views

Good sources to learn about Geometric Analysis

So far the topics that have captured my interest the most are analysis and geometry. I want to learn a bit more about geometric and global analysis. There seems to be no shortage of geometry books but ...
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1answer
56 views

Generalized Laplace--Beltrami operators

Given a smooth surface $M$, we have the well-known Laplace--Beltrami operator which in coordinate-free form is given by $\mathrm{div}_M\,(\mathrm{grad}_M)$. However, we can also consider the more ...
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How to verify F-relatedness?

This question is from Lee's Introduction to Smooth Manifolds p182. I would like to verify the following vector fields are F-related using two ways, i.e. confirming either $dF_p(X_p)=Y_{F(p)}$ for ...
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29 views

Dimension of Conjugacy class in $SU(n)$

Consider $D \in SU(n)$ ($n$ a multiple of 4), a diagonal matrix with values $\pm 1$ on the diagonal and with trace 0 (only possible for $n$ a multiple of 4). There are $n \choose n/2$ such matrices. ...
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Cartan geometry on manifolds with boundary

I was reading Sharpe's text on Cartan geometry, and I started to wonder: Does the theory change in any significant way if the base manifold for the Cartan principal bundle is allowed to be a smooth ...
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22 views

Irreducibility of analytic varieties

Let $V$ be an analytic variety and $V^{*}$ denote the locus of its smooth points. From Griffiths & Harris, page 21, we have that an analytic variety $V$ is irreducible iff $V^{*}$ is connected. ...
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Singular locus of analytic subvarieties

In Griffiths and Harris page 21, it is proven that the singular locus, denoted $V_{s}$ is contained in an analytic subvariety of the complex manifold $M$ not equal to $V$ which is the analytic ...
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1answer
22 views

Connection and curve

Let $\nabla$ be a connection on a Riemannian manifold and let the differential of a curve be given by $$c'(t)=c_1'(t)\partial_1 + c_2'(t) \partial_2.$$ Now I was wondering how we define ...
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1answer
77 views

Convex boundary of a symplectic manifold

Given a symplectic manifold $(M,\omega)$, suppose that $\partial M$ is of contact type. A Liouville field on a symplectic manifold is a vector field $X$ such that $\mathcal L_X \omega = \omega$. We ...
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1answer
27 views

On an informal explanation of the tangent space to a manifold

On Spacetime and Geometry of Sean Carroll pg 17, he states that once a basis is chosen for the tangent space to spacetime at point $p$, say $T_p$, consisting of the vectors ...
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1answer
23 views

Diffeomorphism to tangent space

I had to solve the following problem. Let $M$ be a differenciable $m$-manifold, which admits a global base of differianciable vector fields $\{X_1,\ldots,X_m\}$. This means $\{X_1(p),\ldots,X_m(p)\}$ ...
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26 views

Integral Over the N-Sphere in the framework of chains:

Integration over manifolds is commonly defined with object called chains. What about if I want to integrate the exterior derivative of a $k-form$ over the n-sphere and use Stokes theorem: ...
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Vector identity proof in general curvilinear coordinates, index notation

I need to prove that There is a hint given that I should first lower the index j. I can lower indices with the operation am=Gmjaj . So that what I should do is to multiply both sides of the ...
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1answer
21 views

Supporting lines of closed Jordan curve

Given a simple closed Jordan (i.e. continuous) curve $\gamma:[a,b]\to\mathbb{R}^2,\ \gamma([a,b])=C_\gamma$, how can I prove that $D_\gamma$ (the set of all interior points of $\gamma$ toghether with ...
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64 views

Which spaces admit bump functions?

Let me first fix terminology. Let $X$ be a topological space and $A\subseteq B\subseteq X$ be subsets. Let's say X admits a bump function relative to $A\subseteq B$ if there's a continuous function ...
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25 views

Singer & Thorpe Theorem on the curvature of 4-dimensional Einstein spaces

I recently asked a question here about the paper "The curvature of 4-dimensional Einstein spaces." I got stuck again with the last theorem (2.2), where I get completely lost. They start the proof by ...
4
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4answers
71 views

Can someone illustrate the definition of manifold with a simple example?

In my text the definition of a differential manifold is given as follows: A subset $M$ of $\mathbb{R}^n$ is a $k$ dimensional manifold if $\forall x \in M$ there are open subsets $U$ and $V$ of ...
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1answer
66 views

Differential Equations in Milnor's Topology from the Differential Viewpoint

On page $23$ Milnor states: Let $\varphi$ : $\mathbb{R}^n \rightarrow \mathbb{R}$ be a smooth function which satisfies $$\begin{cases} \varphi(x) > 0, & {\rm for}\,\|x\| < 1 \\ ...