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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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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19 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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25 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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27 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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31 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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13 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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39 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
30 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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76 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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18 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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34 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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32 views

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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26 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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11 views

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
56 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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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
31 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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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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46 views
+100

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
83 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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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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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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42 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
58 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}$ ...
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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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29 views

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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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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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
41 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 ...
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20 views

Translated stochastic process

Let $M$ be a (compact) Riemannian manifold and let $L$ be some second-order elliptic operator on $M$. Now for a vector field $v$, I can consider the flow $\Psi_t$ of $v$ and consider the following ...
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70 views

One parameter subgroup that leaves every compact set is a proper map

If a one parameter subgroup $\phi:\mathbb{R}\rightarrow G$ of a Lie group $G$ comes back infinitely often to a compact set $K$, is it all contained in a compact set? I think $\phi(\mathbb{R})K\subset ...
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26 views

Total variation of real valued functions on a manifold

We can define the total variation of a function $f:\mathbb{R}^n\to \mathbb{R}$ as in Evans and Garriepy's "Measure Theory and Fine Properties of Functions" or in this Wikipedia article ...
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44 views

three vector fields with zero Lie bracket

Suppose I have three unit vector fields $u, v, w$ on a two-dimensional surface in $\mathbb{R}^3$. Are there simple compatibility conditions relating $v$ and $w$ such that there exists scalar fields ...
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1answer
53 views

Gaussian Curvature

I am able to show that if a curve lies in a plane then it's curvature at a point $p$ is $$\kappa=\lim_{\mu\to 0}\frac{\sigma}{\mu}$$ where $\mu$ is the length of a segment of the curve containing p, ...
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53 views

Minimal surface and Weierstraß parametrization

If I have $f(z) = 1$ and $g(z) = \frac{1}{z}$ and I am looking for a minimal surface on $\mathbb{C} \backslash \{0\}$ using the Weierstraß-Enneper representation of minimal surfaces. Now I was ...
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1answer
36 views

Euler characteristic of closed surface

Assume that you have a closed surface that can be covered by finitely many triangles. Then $K(p)= 6-val(P)$ where P is a vertex and $val(P)$ the number of edges that lead to this vertex. Now, I am ...
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2answers
66 views

How long is the curve that a creature walks?

I have a problem in solving mathematical problem. Take a ball with radius 60 cm. A creature walk from the southpole to northpole by following the spiral curve that goes once around the ball every ...
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1answer
48 views

Submanifold associated to blow up.

I 'm trying to understand the classical blow up given by $$X=\{(x,[y])\in \mathbb{R}^n \times \mathbb{P}_N / \hspace{0.2cm} \exists \lambda \in \mathbb{R} \hspace{0.3cm} \text{such that} ...
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+50

General form of a connection with zero curvature

I am looking for proofs of the following two theorems: Theorem 1. On a connected and simply-connected open set $\Omega\subset\mathbb{R}^3$, functions $L^p_{ij}\in C^1(\Omega)$ are given that satisfy ...
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3answers
54 views

normal bundle on a submanifold

Can you give me an example of a nontrivial normal bundle of a submanifold (of any manifold)? There is standard example of the core circle of a mobius band, but can you give an example of a submanifold ...
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1answer
39 views

On a flat surface, can a holonomy can be nontrivial around certain curves

On a flat surface, can a holonomy can be nontrivial around certain curves? How is this possible?
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30 views

Turning number VS winding number

To avoid confusion, here are the definitions of the objects in this question: 1) Let $\gamma:S^1\to\mathbb{R}^2\setminus\{0\}$ a smooth loop. The winding number of $\gamma$ is the number of times ...
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1answer
43 views

Applying Brower's Theorem for Invariance of Domain

I'm working through Manifolds and Differential Geometry by Jeffrey M. Lee. In a topological manifold, we have that every point is in an open set which is homeomorphic to $\mathbb{R}^n$ for some $n$. ...
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143 views

Why $\mathbb{RP}^2$ can not be embedded to $\mathbb{R}^3$?

Is there any answer of this question around basic theory of differentiable manifolds?
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21 views

“Projection of metric” vs. “projection of curvatures”

Suppose we have a submanifold $M^n$ which is embedded in manifold $M^{n+2}$ and $g_{\mu \nu}$ denotes the metric of $M^{n+2}$. We know that the induced metric on the submanifold is defined by ...
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20 views

Topology of operator bundle?

I am trying to understand the family version of the Atiyah-Singer Index Theorem as described in the book "Spin Geometry" by Lawson/Michelsohn. In Part III.§8, they define the operator bundle $$ ...