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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fibration sequence of projective spaces

Question~1: How to construct a fibration sequence $$ S^3\to S^2 \to \mathbb{C}P^\infty\to \mathbb{H}P^\infty ? $$ Does $$S^3\simeq \Omega \mathbb{H}P^\infty ? $$ (Since $\mathbb{C}P^\infty\simeq ...
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Why not differentiable manifolds that are not of class $C^1$

In most, if not all (I cannot say for sure) references on manifolds, we seem to consider $C^k$-manifolds, including the case $k = 0$, which corresponds to topological manifolds. This means that we ...
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Definint the index of a function $f:M\to\Bbb{R}$.

Let $f:M\to\Bbb{R}$ be a function, where $M$ is a $k$-manifold. What will the index at a critical value be? I understand it this way: the only possible critical value of $f$ is $0$. The index at $0$ ...
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2answers
26 views

Is it possible to build a fiber bundle whose fibers are different? (Or we should not call it a fiber bundle?)

Suppose there is a fiber bundle $E$. The base space is $M$ so that $\pi:E\rightarrow M$ is the projection. By the definition, the bundle has a typical fiber $F$ such that the local trivialization over ...
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23 views

Gaussian curvature proof

I can show the first part but not sure how to proceed after that.
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2answers
23 views

Lie Group Automorphism which are diffeomorphism

Is every smooth automorphism of a Lie Group $G$ a diffeomorphism?
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12 views

Surjective $\gamma \colon I \to M^1$, $\gamma (t_1)=\gamma (t_2)$ can be extended to a periodic parametrization of $M^1$

Suppose that $\gamma \colon I \to M^1$ is a smooth surjective curve in a Riemannian connected 1-dimensional manifold. Furthermore, suppose that it is parametrized via arc lenght i.e.:$$||\dot ...
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8 views

Existence of CAT(0)-metrics

Given a metrisable, contractible topological space. Under which conditions exists a CAT(0)-metric compatible with the given topology?
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1answer
25 views

Poincare' s inequality for vectorfields on the sphere

Let $\mathbb{S}^2$ be the standard 2-sphere, and let $V$ be $\mathcal{C}^1$ vectorfield on it. I'd like to understand if it is true that there exists $C > 0$ such that, for all such $V$, we have $$ ...
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12 views

Definition of Linear independence of algebraic $1$-forms

"Suppose that $a,b,c,d$ are linearly independent algebraic $1$-forms on $\mathbb{R}^n$". What does it mean for algebraic $1$-forms to be linearly independent? I have looked through my notes and ...
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20 views

Symmetry of Christoffel symbols of the second kind

I was reading the article: http://physicspages.com/2013/12/22/christoffel-symbols-symmetry/, and I do not understand this: In the locally flat frame, this equation reduces to $\displaystyle ...
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1answer
32 views

tangent space of manifold and Kernel

I want to understand the proof of this Theorem : Theorem: Let $ S $ be a submanifold of $ E $ and let $ a \in S $. Assume that there exists a submersion $f : E \supset U \rightarrow F$ of class $ ...
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Parametrization of a unit 2 sphere

Here is the parametrization for a unit 2 sphere locating at the center of a Euclidean 3 dimensional space: $$x=x(u,v)= \cos u\sin v,\ \ y=y(u,v)=\sin u\sin v,\ \ z=z(u,v)=\cos v, $$ where $0\leq ...
3
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1answer
41 views

Integration by parts (Differential Geometry)

I am studying the proof of a theorem and I am stuck. It says that by integration by parts we get that: For $g(t)$ a variation of Riemannian metrics wih $g'(0)=h,$ $$\int_{M} (-\Delta (tr h) + ...
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13 views

For two unit speed curves $\gamma_{1,2}$ on $[0,l]$ where $0\leq k_2\leq k_1<\pi/l$, show $d(\gamma_1(0),\gamma_1(l))\leq d(\gamma_2(0),\gamma_2(l))$.

For two unit speed curves $\gamma_{1,2}$ on $[0,l]$ where $0\leq k_2\leq k_1<\pi/l$, show $d(\gamma_1(0),\gamma_1(l))\leq d(\gamma_2(0),\gamma_2(l))$. It is important to note here that ...
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1answer
45 views

Inner product, differential forms and surfaces (Stokes' theorem)

I'm trying to understand how do you get the Kelvin-Stokes theorem \begin{equation} \int_{S} (\nabla\times \omega) \cdot \mathrm{d}S = \int_{\partial S} \omega \cdot \mathrm{d}r \end{equation} from the ...
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0answers
20 views

Definition of a connection on a principal bundle

I am trying to understand the definition of a connection as given in, for example, Taubes' book Differential Geometry. Let $\pi: P \to M$ be a principal bundle with a $G$ action. He states that a ...
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1answer
21 views

Counting negative eigenvalues of a Hessian.

Let $f:M\to\Bbb{R}$ be a Morse function. The number of negative eigenvalues of the Hessian at a non-degenerate critical point is the index of $f$ at that critical point. When counting negative ...
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24 views

On the Euler characteristic in Morse Theory

Let $f:M\to \Bbb{R}$ be a Morse function, where $M$ is a $k$-manifold. The index $i_{f,p}$ is defined to be the number of negative eigenvalues of the Hessian $H_f$ at the critical point $p$. For ...
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31 views

Stuck in Preissmann's theorem

I am stuck on following the proof of Preissmann's theorem, whose statement is that Let $(M,g)$ be a closed connected Riemannian manifold of negative sectional curvature. Then every nontrivial ...
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30 views

Angle of intersection between a plane and sphere.

Let $X(\theta,\phi)=(\sin \theta \cos \phi, \sin\theta\sin \phi, \cos\theta)$ be parametrization of the sphere $S^2$. Let $P$ be the plane $x=z \cot\alpha$, $0<\alpha<\pi$ and $\beta$ be the ...
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1answer
42 views

Solvability of system of differential equations

Given $a_i:\mathbb{R}^n \to \mathbb{R}$ $(1\leq i \leq n)$, I am trying to find the conditions under which the equations $$ \frac{\partial f}{\partial x^i}=a_i(x_1,...,x_n) $$ $$ f(x_0)=z_0 $$ is ...
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1answer
14 views

Singular chain complex for integration - pinching on boundary

Singular chain complex, as far as topology are concerned, is just continuous map from standard simplex, and the choice of using simplex over other shape is immaterial. But for integration on manifold, ...
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1answer
41 views

Differential geometry; evaluating the differential $df$ of a function $f$ from the sphere to a meridian and the first fundamental form

Let $C$ be the meridian $C= \{ (x,y,z) \in \Bbb S^2 | y=0,x\geq 0 \}$. Let $f$ map the sphere $\Bbb S^2$ to $C$ such that $f$ maps every point on the sphere to the unique point on $C$ with the same ...
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1answer
31 views

When is a vector field on a manifold restricted to a submanifold $X$ a vector field on $X$?

Let $X$ be an embedded submanifold of $M$ and let $V$ be a vector field on $M$. One can restrict $V$ to $X$, but it may not define a vector field on $X$. Example: The vector field $x^i\partial_i$ on ...
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16 views

Meaning of a Hypersurface resulting from Lagrange Multipliers

Suppose we have a function $f(x_1,\ldots,x_n)$ that we wish to maximize under the set of $n-1$ constrictions $g_i(x_1,\ldots,x_n) = c_i$ for $i \in \{1,\ldots,n-1\}$. We write the Lagrangian ...
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1answer
85 views

Euler characteristic, genus and cohomology: a deep connection?

For a smooth projective curve $V$ over the complex numbers, the algebraic genus, defined as the dimension of the linear system $L(\omega)$, where $\omega$ is the canonical divisor, coincides with the ...
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20 views

Find the lie derivative of a particular integral

Let $\mathbb{X}=(1,y)$ be a vector field on $\mathbb{R}^2$. Let $\Phi_t$ be the flow of $\mathbb{X}$. The flow of $\mathbb{X}$ I have calculated to be $\Phi_t(x,y)=(x+t,ye^t)$ Given a function ...
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38 views

Application Question - American universities strong in Differential Geometry?

Can anyone recommend some American universities (except those top 10 ones such as Harvard, Princeton, SUNY and Umichgan etc. ) which have departments with a solid focus on Geometry and Topology, ...
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Are there some reference books or handbooks on homology and homotopy groups of every manifold which has been calculated?

Are there some reference books or handbooks on homology and homotopy groups of every manifold which has been calculated ? Or Are there some reference books especially on differential geometry and ...
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32 views

Local coordinates on a product of two manifolds.

Let $X, Y$ be two manifolds. Let $(U, x_1, \ldots, x_n)$ and $(V, y_1, \ldots, y_m)$ local coordinates of $X, Y$ respectively. I think that a local coordinate on $X \times Y$ is $(U \times V, x_1 ...
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Poisson bivector on the product of two manifolds

Let $X, Y$ be two manifolds. Let $(U, x_1, \ldots, x_n)$ and $(V, y_1, \ldots, y_m)$ local coordinates of $X, Y$ respectively. A Poisson bivector on $X$ is defined by \begin{align} \pi_X = \sum_{i,j} ...
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1answer
59 views

A difficult question on mathematical physics

Let $TQ^*$ be equipped with its standard symplectic structure and let $X_H$ be a Hamiltonian vector field which is tangent to the fibers of $\pi: TQ^* \to Q.$ I need to show that $$H=h \circ \pi = \pi ...
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1answer
21 views

A Smooth map homotopic to a constant map

Q: Let $M^{k}$ be a smooth compact $k$-manifold and let $F:M \rightarrow S^{n}$ be a smooth map, where $n>k$. Prove that $F$ is homotopic to a constant map. Proof: Since $n>k$, by Sard's ...
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If the linearization of a section is surjective on a slice, is the image under a submersion also a smooth manifold?

Let $V \rightarrow M \times N_1 $ be a vector bundle, where $M$ and $N_1$ are smooth manifolds and $s: M \times N_1 \rightarrow V$ a smooth section such that if $s(p,q) =0$ then $$ \nabla ...
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1answer
32 views

Topological structure of the Manifold valued functions

$M$ is a Riemannian manifold. What condition on $M$ for $\mathcal{C}_{[a,b]}(M)$ (the set of continuous functions of the real interval $I=[a,b]$ to $M$) to be a polish space ? For which topology ? Is ...
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1answer
28 views

How to prove line bundle L is trivial if and only if its dual bundle us trivial?

How to prove line bundle L is trivial if and only if its dual bundle us trivial ?
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64 views
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Proof of holomorphic Lefschetz fixed point formula using currents in Griffiths and Harris

I am trying to understand the proof of the Holomorphic Lefschetz fixed point formula on page 426 in Griffiths and Harris. However, I find their use of currents extremely confusing. They seem to go ...
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3answers
84 views

Can a non-zero vector field have zero divergence and zero curl?

I don't see how. Curl and divergence are essentially "opposites" - essentially two "orthogonal" concepts. The entire field should be able to be broken into a curl component and a divergence component ...
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1answer
53 views

Is (0,0) of $V(x-y^2)$ a smooth point?

I'm pretty sure it is a smooth point since given $f(x,y)=x-y^2$ the gradient $df=(1,-2y)$ is always non-singular. I'm asking because page 22 of Principle of Algebraic Geometry says: ...
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1answer
47 views

Weak tangent but not a strong tangent

Question: Show that $\alpha(t)=(t^3,t^2)$, $t\in \Bbb R$, has a weak tangent but not a strong tangent at $t=0$. Definitions from this answer: (Weak tangent) $\alpha: I \to \Bbb R^3$ has a weak ...
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1answer
43 views

Restricting the DeRham cohomology class of a submanifold to a coordinate neighborhood.

Suppose $M$ is an $n$-manifold and $A$ a $k$-dimensional submanifold, both compact and oriented. Let the deRham cohomology class of $A$ be denoted $[\phi_A]$. The class is defined by ...
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2answers
60 views

Definition of a Manifold from Guillimen Pollack

I have been studying differential topology from Guillimen and Pollack (GP). Unlike many other books that define differentiable manifolds using maximal atlases GP starts by saying $ X \subset R^{N}$ ...
3
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1answer
131 views

Confusing Analysis proof

I have a question about a proof of the Beltrami-Enneper theorem: In the following $\nu$ is the surface-normal and $e_1,e_2,e_3$ the Frenet 3-frame. It states: Every asymptotic curve $c: I \rightarrow ...
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0answers
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Parameterization of surface of revolution with constant mean curvature

Let $x(u,v) = (g(u), h(u) \cos v, h(u) \sin v)$ be a parameterization of a surface of revolution $M$, arising from rotating the regular curve $\alpha(u) = (g(u),h(u),0)$ around the $x$-axis with ...
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0answers
41 views

How to define an action of vector field on $C^{\infty}(M)$?

Let $M$ be a manifold. Let $\hat{X}$ be a vector field on $M$. How to define an action of $\hat{X}$ on $C^{\infty}(M)$? Thank you very much.
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1answer
25 views

How to define the vector field $\hat{X}$ on a manifold $M$ defined by an element $X$ in a Lie algebra $\mathfrak{g}$?

I read a paper and on page 9, the paragraph before Proposition 5, it is said that let the vector field $\hat{X}$ on a manifold $M$ defined by an element $X$ in a Lie algebra $\mathfrak{g}$. How to ...
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1answer
54 views

Differentiating with respect to a vector

Hello i'm new to this forum and this is my first post. I was going over the transport theorems in fluid mechanics and there is one way in which you can convert reynolds transport theorem into a single ...
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2answers
45 views

Showing that a specific curve is regular.

Define the curve by $c(t):=(sin(pt)+r)(cos(qt),sin(qt))$ for $p,q \in \mathbb{Q}$ and $r\in \mathbb{R}$. Determine for which $p,q$ is the curve regular, i.e. $c'(t) \neq (0,0)$ for any $t\in ...
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1answer
42 views

Equivalent definitions of Euler characteristic for closed manifolds

It is well-known that the Euler characteristic of a closed manifold $M^n$, which can be defined as $\chi(M)=\sum_{k=0}^n (-1)^k \operatorname{dim}H^k(M)$, equals the intersection number ...