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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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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74 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
25 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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25 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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34 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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24 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
18 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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42 views

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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40 views

Is every regular curve homeomorphic to an interval I $\subset \mathbb{R}$ or to $\mathbb{S}^1$ or are there other posibilities? [on hold]

Is every regular curve always homeomorphic to an interval or to $\mathbb{S}^1$? If so I would like to know why.
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1answer
15 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 finite. (So $G$ is defined on a ...
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26 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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12 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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21 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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42 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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52 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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14 views

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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28 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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19 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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17 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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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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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
20 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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25 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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7 views

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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18 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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62 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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22 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 ...
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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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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 \\ ...
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1answer
19 views

Locality of tensors part of definition?

I am wondering whether linearity with respect to scalar functions $f \in C^{\infty}(M, \mathbb{R})$ is part of the definition of a tensor? Let me explain it by referring to the Riemann curvature ...
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25 views

Finding Riemannian metric from this geodesic

In a $d$-dimensional Riemannian manifold, given a geodesic equation $\gamma^i(t)=a^i\phi(tb^i),i\in 1\ldots d$, where $\phi:\mathbb{R}\rightarrow\mathbb{R}$ is an increasing function, $a^i,b^i$ are ...
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Show that $T(t)$ and $N(t)$ are Orthogonal

If $r(t)$ is the smooth parametrization of a curve $C$ in 3-space, then the unit tangent and unit normal vectors are denoted as $T$ and $N$, and are given by: ...
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27 views

it is possible to define a topology on particular vector space

If i take $V$ a finte dimensional vector space on the real number (or complex number). Setting $n=dim_{\mathbb{R}}(V)$, i know that there is a isomorphism of vector spaces so $V \simeq \mathbb{R}^n$. ...
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question about how to make a set a differential variety

If i take $M$ a differential real manifold o finite dimension $n$ at every point of $M$, $m_0$, i can construct the tangent space in $m_0$ to the variety $M$ named $T_{m_0}M$. So this space is a real ...
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30 views

Topological properties of regular and critical points and values

Let $f\colon M\rightarrow N$ be a smooth map between smooth manifolds. Consider the following two statements, the second one under the assumption The set of regular points of $f$ are open in $M$, ...
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52 views

Parallel Transport on a Cone

Suppose we have a cone and we wish to parallel transport a vector $w=(0,1,0)$ from along the curve $\alpha(s)=(\sqrt{2}/2 \cos(v\sqrt{2}),\sqrt{2}/2 \sin(v\sqrt{2}),\sqrt{2}/2)$ from $p=\alpha(0)$ to ...
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2answers
27 views

Proving Hadamard lemma: how to apply FTC in first step?

I wanted to prove Hadamard's lemma but got stuck on the first step: Let $f \in C^\infty (\mathbb R^n)$ and $x_0 \in \mathbb R^n$. Then there exist $g_i \in C^\infty (\mathbb R^n)$ such that $$ f(x) ...
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Checking the proof: Ric=Kg

Let $(M,g)$ be a semi-riemaniann surface (dim M =2). Let Ric be the Ricci tensor and K the sectional curvature. Then Ric = Kg. Proof: Let $R$ be the Riemann curvature of $(M,g)$. Let $x, y \in T_pM$ ...
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Ricci flow on surfaces : step in proof

I am trying to realize the paper of richard hamilton's ricci flow on surfaces from the book of benett chow's Ricci flow : An Introduction.Here Hamilton denoted the trace free part of the Hessian of ...
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1answer
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Orthogonal tangent vectors on Sphere in arbitary dimension

Consider $\mathbb R^{4}$ with coordinates $x^{1},...,x^{4}$. We can write down the following forms: $$x^{1}dx^{2}-x^{2}dx^{1}+x^{3}dx^{4}-x^{4}dx^{3}$$ ...
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44 views

Tangent map of orthogonal projection

Given a tangent bundle $TM$ and its natural projection $\pi:TM\to M$, I want to compute tangent map $T\pi:TTM\to TM$. Here is my method. Suppose a curve $c:\mathbb{R}\to TM$ with $c(0)=(x,y)$ and ...
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38 views

Are $\mathbb{CP}^{n}$ and $\mathbb{RP}^{2n}$ diffeomorphic?

I understand that they are homeomorphic but couldn't find a proof that they are diffeomorphic. If they are diffeomorphic and if the proof is simple enough, I would imagine it would look like the ...
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13 views

Projection of vector onto hyperplanes

I have some questions about the following excerpt from the paper "Constrained Shrinking Dimer Dynamics for saddle point search with constraints." by Zhang and Du. Let $G(x)=(G^1(x),G^2(x), ...
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Is it necessary for $A$ to be symmetric with non-zero determinant?

Today, in a Differential Geometry test, I was asked to prove that: $$S:=\{x \in \mathbb{R}^3: x^TAx+c=0\}$$ where $A$ is a symmetric $3\times 3$ matrix and $c \in \mathbb{R}$ is a regular surface ...
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What is the space $T_Y^*X$?

Help with notation! I'm a physicist, but I've come across the following notation, $T_Y^* X$ where $Y$ is a complex analytic submanifold of $X$. A phrase I've heard used is "conormal bundle" but is ...