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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Is there any situation in which a geodesic maximize the path length between two points?

Some people (even in here) claim that geodesics are, in general, stationary curves. Locally speaking, geodesics always minimize arc length (see Manfredo, for example). But I can't visualize a surface ...
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17 views

If $0 < \theta < \frac{\pi}{2}$, then $\gamma$ is a logarithmic spiral

Let $\gamma$ be a plane curve parametrized by the arc length, having the property that its tangent vector $T(t)$ forms a fixed angle $\theta$ with $\gamma(t)$. Before explaining where I am, let ...
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16 views

Parametrization where coordinates lines are lines of curvature

I am asked to prove that given a surface $S$ and a point $p\in S$ non-umbilical, then there exists $U$ open in $\mathbb{R}^2$, there exists $Y:U\subset \mathbb{R}^2\longrightarrow \mathbb{R}^3$ a ...
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1answer
49 views

Does there exist a harmonic map from S^2 to 3d hyperbolic space

My question is, does there exist a harmonic map from $S^2$ to $\mathbb{H}^3$ , $\mathbb{H}^3$ means the 3d hyperbolic space. In addition, if it exist, could we directly construct the map? Thank you ...
3
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34 views

Proving existence of local diffeomorphism

Consider the setup from here: Do these vector fields span an integrable distribution? For any pair of points $p, q \in U$, show that there is a local diffeomorphism $F: U(p) \to U(q)$, such that ...
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26 views

the gradient in complex coordinates

Let $M$ be an $n$-dimensional complex manifold, or equivalently a $2n$ real manifold. Let $g$ be a Riemannian metric. Let $f\in C^\infty(M)$. What is $\nabla f$? If $x_1,\dots,x_n,y_1,\dots,y_n$ are ...
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1answer
17 views

Relationship between $\operatorname{supp} w$ and $\operatorname{supp} dw$

I would like to show that $\operatorname{supp} dw\subset \operatorname{supp}w$, where differential $m$-form in a $\Sigma$ surface of dimension $m+1$. If $$w(u)=\Sigma_i (-1)^ia_i(u) du_1\wedge ...
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1answer
60 views

Why don't all metrics have trivial determinant?

What is wrong with this argument? Let $V$ be a vector space and $g$ an inner product. There exists an orthonormal basis for $V$. That is, in this basis $(g_{ij})=I$. But then given any other basis, ...
3
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2answers
40 views

Do these vector fields span an integrable distribution?

Let $X_i$, $1 \le 1 \le n$, be smooth vector fields on the open set $U \subset \mathbb{R}^n$, which are linearly independent at each point. Set $[X_i, X_j] = \sum_{k=1}^n c_{ij}^kX_k$. Suppose that ...
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11 views

Adjacent angle Theorem in planes of constant curvature - easy geometric proof?

I am trying to proof the Adjacent angle Theorem in planes of constant curvature (2-Sphere, euclidean plane, hyperbolic plane) i.e given 4 points $a,b,c,d$ such that $d$ is lying on a shortest curve ...
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26 views

Relative version of Whitney embedding's theorem (reference needed)

One form of Whitney embedding theorem says that if $M,N$ are smooth compact manifolds and the dimension of $N$ is more than twice the dimension of $M$, then the space of embeddings $M\to N$ is open ...
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1answer
40 views

Differential Geometry-Wedge product

How can we prove the following relation for differentiating the wedge product of a p-form $\alpha_p$ and a q-form $\beta_q$$$d(\alpha_p\wedge\beta _q)=d\alpha_p\wedge\beta_q+(-1)^{p}\alpha_p\wedge ...
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47 views

Identifying the cotangent bundle of the flag variety

Suppose $G$ is a Lie group (or I guess a linear algebraic group), $P \subset G$ a Lie subgroup with Lie algebras $\mathfrak{g}$ and $\mathfrak{p}$ respectively. In Chriss and Ginzburg's book ...
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37 views

Curvature and Pfaffian forms in terms of the Riemann tensor

I am teaching my self differential geometry, but I am mainly familiar with classic tensor notation. In modern Cartan exterior form notation the curvature forn $\Omega$ and the Pfaffian seem to do the ...
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1answer
37 views

Showing that a $X(u,v)$ is one-to-one

I am currently reviewing for a differential geometry exam, and seem to be having trouble with the algebra required to show that the parameterization $$X(u,v)=(u+v, u+v, uv)$$ is one-to-one. I know ...
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1answer
61 views

Elementary tensors [duplicate]

I need to determine whether the following function is tensor on $\Bbb R^4$ and express it in terms of elementary tensors. Can someone please help me with it? I do not know what elementary tensor means ...
2
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0answers
45 views

Continuity of maximum distance between geodesics on a smooth manifold

I am working on my own version of a proof of the Jordan Separation Theorem (just for fun - I know it's been proved countless times) and in the course of so doing I use the apparently fairly obvious ...
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0answers
22 views

Uniform convergence of the harmonic form heat flow

[${\bf NOTATIONS}$] Let $M$ be a closed Riemannian manifold of $m$ dimensional, $p\in\{1,\cdots,m\}$. $A^p:=\{\text{smooth p-forms on }M\}$. $\delta:A^{p+1}\to A^p$ denotes the formally adjoint ...
1
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1answer
27 views

If differential forms agree on one chart do they agree everywhere?

Let $\alpha,\beta$ be two $k$-forms on a manifold $M$. If there exists some chart $(U,x^1,\dots,x^n)$ on which $\alpha=\beta$ does it follow that $\alpha$ and $\beta$ are the same forms? In different ...
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0answers
19 views

Moment map and Hamiltonian

Take the manifold $M$ to be $M=\mathbb{R}^6=\mathbb{R}^3\times\mathbb{R}^3$ (hence $x\in M$ is given by $x=(p,q)$ with $p$ and $q$ three dimensional vectors) and take the possion bracket on $M$ given ...
2
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1answer
48 views

Critical Points of a smooth map on SO(n)

I am given the following map $f:SO(n) \rightarrow \mathbb{R}$, $f(X) = Tr(DX)$ where $D$ is a diagonal matrix $\{d_1,\ldots,d_n\}$, $1<d_1<\cdots <d_n$. I need to find the critical points ...
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2answers
1k views

Apparent counter example to Stoke's theorem?

I think I found an apparent contradiction to Stoke's theorem with this 2-differential form $M= \overline{B^{2}}- \{ 0 \}$, $\partial M = S^1$, $$\omega = \frac{x~dy-y~dx}{x^2+y^2}$$ defined in ...
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1answer
19 views

Kernel of a Constant Rank Bundle Map Using the Constant Rank Theorem

Let $M$ be a smooth manifold (without boundary) and $E$ and $E'$ be smooth vector bundles of over $M$. Let $F:E\to E'$ be a bundle homomorphism. For each $p\in M$, we define the rank of $F$ at $p$ ...
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26 views

Svarc-Milnor Lemma to prove that finite index subgroup of f.g. group is finitely generated

I found a proof using Švarc-Milnor lemma (the Lemma is prop. 1.19 here) of the well known fact that a subgroup of finite index of a finitely generated group is finitely generated (here a proof with ...
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1answer
29 views

Totally geodesic submanifold

I'm reading "Introduction to symplectic topology", D.McDuff, D.Salamonand and I have a problem with the exercise 1.26. According to the definition, a submanifold $L$ of a Riemannian manifold $(\mathbb ...
2
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0answers
26 views

How do the differentials you get from complex functions behave?

I'm starting complex analysis and I've learned about the required conditions for a complex valued function of complex variables to be derivable. For the closest real analog, the $\mathbb R^2 ...
0
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1answer
64 views

Connections and tensor fields

Let $T$ be a $(1, 1)$ tensor field, $\lambda$ a covector field and $X, Y$ vector fields. We may define $\nabla_X T$ by requiring the ‘inner’ Leibniz rule, $$\nabla_X[T(\lambda, Y )] = ...
2
votes
1answer
19 views

Hodge star related question

If we have expression (1) $$\star (F \wedge d\alpha)$$ where $$ (F \wedge d\alpha)$$ is a $2$-form field strength ($F$ and $d\alpha$ are 1 forms)and $\star$ represents Hodge star. How can we ...
3
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1answer
76 views

Geometrical interpretion of signed curvature

What does signed curvature mean geometrically? What is the difference between curvature and signed curvature? How i can determine signed curvature of curve from the graph of that curve?
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15 views

a vector space with indefinite scalar product [on hold]

The Problem of (SEMI-RIEMANNIAN GEOMETRY WITH APPLICATIONS TO RELATIVITY BARRETT O'NEILL) section 7, page 232, 9. In a vector space with indefinite scalar product, every degenerate 2-plane is a limit ...
4
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1answer
40 views

Lifting an $\mathbb{RP}^{n-1}$-valued map to an $S^{n-1}$-valued map.

Let $M$ be a smooth manifold and assume that for each $v\in M$, I have a vector $v(m)\in\mathbb{R}\setminus\{0\}$ with some property and this vector is unique up to multiple: further, it depends ...
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0answers
57 views

Riemannian metric induced by metric

This seems a very basic and useful construction, and yet I cannot find any reference for it. So my questions are, 1) Is the following definition correct? 2) Is there a simpler construction? 3) Do you ...
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1answer
34 views

Two isometries that have same value and differential at some point are the same.

I also have trouble in this problem: Let $f, g$ be two isometries of a connected Riemannian manifold $(M, g)$. If $f(p)=g(p)$, $df_p = dg_p$, show that $f=g$. Any comment is expected. I know it ...
0
votes
1answer
19 views

Minimal geodesic on the real projective $n$-space

I have encountered this problem: Show that a geodesic of $(\mathbb RP_n, g_0)$ with $g_0$ being the metric given by the canonical metric on $\mathbb S^n$ via the $2:1$ Riemainnian covering, is ...
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0answers
40 views

Manifold and the topology of $\mathbb{R}^{n}$

A manifold $M$ is defined in particular as being locally homeomorphic to $\mathbb{R}^{n}$. Homeomorphisms can be defined in terms of how they map open sets, namely an homemorphism $f$ and its inverse ...
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1answer
37 views

Frame acting on a curve/Geodesic eqution

I have a technical question about the geodesic equation. Assume we have a frame $(E_{1},E_{2},E_{3},E_{4})$ (not necessarily a coordinate frame). Assume we have a parametrized curve $\gamma(s)\in M$ ...
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0answers
52 views

Integration on $ \mathbb{P}^n ( \mathbb{R} ) $.

Could you tell me please, when and how we calculate the integral of a function on $ \mathbb{P}^n ( \mathbb{R} ) $ ? Do you have some references about that ? Thank you in advance.
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0answers
15 views

Question about uniform wire and its application to find centroid

A uniform wire has the shape of that portion of the curve of intersection of the two surfaces x^2+y^2=z^2 and y^2=x connecting the points (0.0.0) and (1.1.square root 2) Find the z-coordinate of its ...
1
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1answer
48 views

$M=\{(x,y,z) \in \mathbb{R}^3 : xyz=C\}$ is a manifold $\Rightarrow C \neq 0$

I'm having some troubles on showing that $M=\{(x,y,z) \in \mathbb{R}^3 : xyz=C\}$ is a manifold $\Leftrightarrow C \neq 0$ I have already proved $\Leftarrow$ but I can't see how to prove ...
1
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3answers
59 views

What means that two manifolds have “the same topology”?

I know this is a very basic question, but let me be more specific. Suppose that, for definiteness, $M$ and $N$ are differentiable manifolds. What means that they have the same topology? Does this mean ...
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2answers
41 views

Hyperbolic geometry when the curvature is constant and negative but not -1

Help I am getting completely confused Hyperbolic geometry is the geometry of surfaces of a constant negative Gaussian curvature, in most formula's it is almost assumed this constant negative ...
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20 views

Uniform semicircular wire of radius “a” and its application [on hold]

Consider a uniform semicircular wire of radius a a) show that the centroid lies on the axis of symmetry at a distance $2a/\pi$ from the center. b) show that the moment of inertia about the diameter ...
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2answers
43 views

$GL(n,K)$ is open in $M(n,K)$

I want to prove, that $GL(n,K)$ is open in $M(n,K)$, where ($K=\{\mathbb{R},\mathbb{C},\mathbb{H}\}$. For the prove I don't want to use that the determinant is continous. Alternativly I assume a ...
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1answer
23 views

Basis for curvillinear coordinate systems.

I am reading through Geometry of Physics - Frankel and in the preface of the latest edition, Frankel defines a curve $C_i$ through a point $p$ parametrized by $u^j=constant$, $j\neq i$, with ...
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2answers
45 views

Poincaré–Bendixson theorem

Does someone know a good reference for a proof of the Poincaré–Bendixson theorem using the language of vector fields?
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1answer
23 views

preservation of the curvature tensor implies preservation of the connection?

For every connection on a smooth manifold there is a corresponding curvature tensor. Any diffeomorphism $\phi:(M,\nabla^M)\rightarrow(N,\nabla^N)$ which preserves the connection (in the sense of ...
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0answers
21 views

Differentials as basis vectors [closed]

I am reading through Geometry of Physics - Frankel and in the preface of the latest edition, Frankel defines a curve $C_i$ through a point $p$ parametrized by $u^j=constant$, $j\neq i$, with ...
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1answer
36 views

Does this set of symmetric matrices form a smooth manifold?

Let $A$ be a real symmetric $n \times n$ matrix. Let $1 \leq i < j \leq n$ and $1 \leq k < \ell \leq n$. Let $A'$ be the submatrix of $A$ consisting of rows $i,...,j$ and columns $k,...,\ell$. ...
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0answers
41 views

Pre-requisites for studying differential geometry?

I am an 3rd year undergrad interested in mathematics.i had read h.graham flengg (from geometry to topology).i found this field interesting i now want to read further,so i want to ask these questions ...
3
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3answers
57 views

Finding Gauss curvature of surface

Consider the surface $S=F(\mathbb{R}^2)$ where $F:\mathbb{R}^2 \to \mathbb{R}^3$ is defined by $$(r, \varphi) \mapsto ( r \cos \varphi, r \sin \varphi, \varphi).$$ I would like to find the Gauss ...