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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Equation of a straight line in spherical coordinates

I'm trying to prove the angle sum formula for a triangle on the surface of a sphere. In order to do this I wanted to create a general triangle on the sphere, with one vertex at $\theta = 0$ and one ...
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1answer
20 views

Distance in the $y$-axis of the hyperbolic plane

I'm reading Stillwell's Geometry of Surfaces but I'm having a little bit of trouble because my background in calculus isn't great, I'm struggling with these problems: In the upper half-plane model, ...
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1answer
81 views

Multiplying two tensors of the Levi-Civita type

How to multiply two epsilons with one another? We know ...
2
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1answer
70 views

Hausdorff Dimension of a manifold of dimension n?

Let's say that $M$ is a differentiable manifold of dimension $n$. (This includes that $M$ is nonempty and second countable, so that it can be embedded into some Euclidean space, and is thus ...
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1answer
375 views

Can we generalize the regular value theorem even beyond the Ehresmann's theorem?

The formulation is complicated, but the answer may be some clever usage of the partition of unity, because locally the answer is given by the regular value theorem and the whole problem is to glue it ...
3
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1answer
85 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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1answer
315 views

Stokes theorem on Lipschitz-manifolds?

I was wondering if Stokes' theorem could be formulated in a setting which could be easily applied in situations where the traditional form cannot, such as on manifolds with corners like a rectangle or ...
4
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0answers
52 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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20 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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25 views

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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15 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
30 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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29 views

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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24 views

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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21 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
132 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 ...
2
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0answers
23 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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1answer
25 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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1answer
37 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
26 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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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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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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1answer
34 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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1answer
50 views

Diameter of the Grassmannian

Just an interesting question that came to my mind while studying(!): Since the Grassmannian $G(k,\mathbb{C}^n)$ is a compact manifold, what do we know about its diameter? Do we know any estimate? ...
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1answer
81 views

Ricci curvature of the Grassmannian?

Let $G(k, \mathbb{C}^n)$ be the Grassmannian of $k-$dimensional complex linear subspaces of $\mathbb{C}^n.$ We know that the Grassmannian can be embedded to the projective space ...
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1answer
84 views

Grassmannian a Fano manifold?

I am interested in answering the following: Is it true that the Grassmannian $G(k,\mathbb{C}^n)$ is a Fano manifold? How can I see if its anticanonical bundle is ample?
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1answer
103 views

Perturb a given smooth function to a Morse function relative to fixed level sets, which are already fine.

Let $M$ be a manifold (not necessarily compact) , for the sake of clearness embedded in $\mathbb{R^n}$ and $f\colon M\rightarrow \mathbb{R}$ a smooth function. The theorem of Sard gives us that ...
6
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1answer
62 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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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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0answers
41 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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1answer
22 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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30 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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0answers
38 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 ...
7
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1answer
86 views

Motivation for the definition of an infinitesimal object

An infinitesimal object $D$ in a Cartesian closed category $\mathsf{C}$ is one for which the internal Hom functor $$(-)^D: \mathsf{C} \to \mathsf{C}$$ has a right adjoint. I am wondering what is the ...
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0answers
104 views

Some questions about synthetic differential geometry

I've been trying to read Kock's text on synthetic differential geometry but I am getting a bit confused. For example, what does it mean to "interpret set theory in a topos"? What is a model of a ...
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1answer
43 views

reconstruct space curve from $\kappa=\frac{a}{a^2+b^2}$ and $\tau=\frac{b}{a^2+b^2}$

I have a problem to solve, and i have a solution, but not sure if it is right one. E.g. i have never used that $a > 0.$ Can you please look at it and correct it if something went wrong. Thanks. ...
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0answers
15 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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2answers
35 views

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
25 views

Describe a parallel transport on sphere [on hold]

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 ...
4
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1answer
50 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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0answers
30 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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36 views

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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1answer
89 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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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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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 ...
2
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0answers
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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18 views

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
29 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
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 ...