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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Example for divisors, line bundles and meromorphic functions on $\mathbb{CP}^2$

I have been studying divisors using Griffiths/Harris (chapter 1.1) as well as Huybrechts (chapter 2.3). However, I cannot seem to find any very easy worked examples - i.e. $\mathbb{CP}^1$ or ...
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Equivalent descriptions of “flat space with non Euclidean metric” and “curved space with (local) Euclidean metric”: the case of Minkowski space.

FIRST: I start with the guiding idea: 1. we have the surface of a paraboloid (z = x2 + y2); its metric, in an infinitesimal neighbourhood of one of its points is (we can choose it) EUCLIDEAN; now, ...
2
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1answer
42 views

Riemannian metric and geodesic completeness

For all $p \in \mathbb{R} \smallsetminus \lbrace 0,1 \rbrace$, let $\displaystyle M(p)=\frac{1}{p^{2}(p-1)^{2}}$. Then, let $g_{p} \, : \, (u,v) \, \longmapsto \, uM(p)v$. I am not sure about the ...
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1answer
21 views

When do isometries commute with the compatible derivative operator on a semi-Riemannian manifold?

Let $M$ and $\tilde{M}$ be smooth manifolds, each with a metric $g_{ab}$ and $\tilde{g}_{ab}$, assumed here to be smooth symmetric invertible tensor fields, which are non-degenerate but not ...
2
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1answer
26 views

Differentiation of scalar fields using tensor notation

I'm learning tensor calculus to understand differential geometry. Please verify if I've understood how to employ Einstein's sum convention and index notation correctly. Suppose that $\varphi := ...
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24 views

Interpretation of $(r,s)$ tensor

A tensor of type $(r,s)$ on a vector space $V$ is a $C$-valued function $T$ on $V×V×...×V×W×W×...×W$ (there are $r$ $V$'s and $s$ $W$'s in which $W$ is the dual space of $V$) which is linear in each ...
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1answer
85 views

Example of Something That's Not A Manifold

Two examples of non-manifolds that I know are the cross and the cone. Also the sphere with a hair isn't a topological manifold. But what's an example of a topological space $X$ such that $X$ is not a ...
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1answer
36 views

Is $\mathbb{R}P^n$ “two-sided” in $\mathbb{R}P^{n+1}$

i.e. Does $\mathbb{R}P^n$ have a tubular neighborhood $N$ such that $N-\mathbb{R}P^n$ is disconnected. My guess is yes, but don't know how to show it convincingly ( or maybe only for $n$ odd, I'm ...
4
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37 views

Differential forms and determinants

2-forms are defined as $du^{j} \wedge du^{k}(v,w) = v^{j}w^{k}-v^{k}w^{j} = \begin{vmatrix} du^{j}(v) & du^{j}(w) \\ du^{k}(v) & du^{k}(w) \end{vmatrix}$ But what if I have two concret ...
2
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2answers
46 views

Ricci Tensor, Curvature and Scalar Curvature computation from definition

I am studying a little of Riemannian geometry and I am having some problem in making the connection between two expressions of Ricci tensor, curvature and scalar curvature. Well, in the book that I am ...
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31 views

Manifold has uncountable many smooth stuctures if it has one

This is the Problem 1-6 of John Lee's Introduction to smooth manifold: Let $M$ be a nonempty topological manifold of dimension $n\geq1$. If $M$ has a smooth structure, show that it has uncountably ...
0
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0answers
32 views

Sequence of differential forms.

My question is: how can I prove for the sequence of differential 1-forms $\{\omega_{k}\}_{k}$ that $\{d\ast d \omega_{k}\}_{k}$ is contained in some compact subset of the space $L^{2}(\Omega)$ (in ...
2
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1answer
25 views

Vanishing pushforward implies smooth function is locally constant?

I'm trying to prove that if the pushforward $dF$ of a smooth map $F\colon M\to N$ between smooth manifolds is zero, then $F$ is constant on each component. It will be enough to show $F$ is locally ...
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1answer
17 views

Inverse Image of a Regular Value an Orientable Submanifold

Let $f:M^n \rightarrow \mathbb{R}$ be a smooth map, and let $c\in N$ be a regular value. When is $f^{-1}(c)$ an orientable manifold? Note: I know by regular value thm, $f^{-1}(c)$ is a smooth $n-1$ ...
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1answer
27 views

Orthonormal basis for a tangent plane

Given a manifold $M$ described by the graph of an arbitrary smooth function $f:U \subset \mathbb{R}^2 \to \mathbb{R}^3$, I would like to construct an orthonormal basis for its tangent plane $T_pM$ at ...
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0answers
33 views

there exists a unique plane in a point of a surface in $\mathbb{R}^3$ [on hold]

The question is how I can prove the existence in this problem: If $M\subset \mathbb{R}^3 $ is a smooth surface. Then, there exists a unique plane $\Gamma\subset \mathbb{R}^3$ that passes through ...
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0answers
24 views

Paraboloid Curvature calculation methods

If we have a paraboloid generated as a surface of revolution of the 2d function $f(x)=ax^2+b$, the equation of the 3d graph is $f(x,y)=ax^2 + ay^2+b$. The gaussian curvature of a 3d graph $f(x,y)$ is ...
2
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3answers
235 views

Holomorphic functions on a complex compact manifold are only constants

Is there a simple proof that every holomorphic function $M\to\mathbb{C}$ on a compact complex manifold $M$ is constant?
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1answer
23 views

Estimate for boundary points and exterior normal vector of bounded domain of class $C^2$

Consider a bounded open set $\Omega\subset\mathbb{R}^d$, s.t. the boundary set $\partial \Omega$ is a manifold of class $C^2$. Let $x,x_0\in\partial\Omega$ be boundary points and $\nu_x$ the exterior ...
3
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0answers
28 views

Doubt with smooth extensions

Let $(x,y)\in\mathbb R^2$ and $M$ a manifold defined by $M=\left\{ (x,y)\in\mathbb R^2\, |\, y^2+x=0 \right\}$. Let $\pi$ be a projection $\pi(x,y)=(x)$. Let $\phi:\mathbb R\to\mathbb R$ be a ...
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0answers
37 views

Parallel Transport of Geodesic Velocity Vectors

Given a Riemannian manifold $M$ with Riemannian metric $g_{x}:T_{x}M\times T_{x}M\rightarrow\mathbb{R}$ and distance $d:M\times M\rightarrow\mathbb{R}$ determined by length of minimizing geodesics, ...
3
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1answer
54 views

What's the Differential of this Map $f:S^3\rightarrow \mathbb{R}$

$f:S^3 \rightarrow \mathbb{R}$ is defined as $f(x,y,z,w)=x+zw$, where $S^3= \{(x,y,z,w) | x^2 +y^2 +z^2 +w^2 =1\}$ I tried using a stereographic chart but that got ugly. The function is so simple I ...
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1answer
106 views
+400

Non-ellipticity of Yang-Mills equations

Let $D=\text{d}+A$ be a metric connection on a vector bundle $E\rightarrow M$ with curvature $F=F_D$. How does one prove that the Yang-Mills equations $$ \frac{\partial}{\partial ...
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1answer
36 views

Levi Civita connection along principal curvature directions

Let $(M,g)$ be a surface that can be immersed into $\mathbb{R}^3$. Denote by $\nabla$ the associated Levi Civita connection. Further, let $X_1,X_2$ be the directions of principal curvature which are ...
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0answers
30 views

Question on theorem: $\delta_{g}F(x)=0\Leftrightarrow\left[g,F\right]=0$ [on hold]

Can anyone explain me why the following statements are the same? $$\delta_{g}F(x)=0\Leftrightarrow\left[g,F\right]=0$$ Is there any theorem with this statement? Thanks in advance!
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0answers
25 views

Connection defined by its geodesics

I have a question related to the definition of a connection (in the sense of Koszul) by its geodesics. I know that a torsion free connection is uniquely determined by its geodesics. Now, let $M$ be a ...
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1answer
44 views

Is a level set of a manifold a set of zeroes

Suppose $X$ and $Y$ are manifolds of dimensions $k$ and $l$ (with $k>l$). Given $F : X \to Y$ a smooth map and $y$ a regular value in $Y$, does there exist a map $G : X \to \mathbb{R}^l$ such that ...
0
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1answer
15 views

Reducible to Separable First Order Differential Equation Word Problem in Analytic Geometry 1.4-29

I completed near all problems om a differential equations text chapter on reducing non-separable first order differential equations to separable by using an appropriate substitution for example u = ...
1
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1answer
14 views

Reducible to Separable First Order Differential Equation Word Problem in Analytic Geometry 1.4-28

I completed near all problems of a differential equations text chapter on reducing non-separable first order differential equations to separable by using an appropriate substitution for example $u = ...
1
vote
2answers
96 views

Non existence of a non singular vector field on $S^2$

Prove that the unit tangent bundle of $S^2$, $T^1 S^2$, is not diffeomorphic to $S^2×S^1$ by showing that if so there exists a nowhere vanishing vector field on $S^2$ I do not know how to create that ...
9
votes
2answers
141 views

Is there a retraction of a non-orientable manifold to its boundary?

It's easy to show using Stokes theorem that a compact orientable manifold with boundary cannot retract to its boundary, by choosing a volume form. But for the non-orientable case I don't know if this ...
3
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0answers
31 views

Second fundamental form of a graph of a function using frame fields

I am trying to recover the second fundamental form $II$ for the graph $\Gamma(f)$ of a smooth real function $f$ in the Euclidean space $\mathbb{R}^3$ using first order frame fields. I have worked with ...
2
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0answers
34 views

Connections on principal bundles and vector bundles

In Donaldson and Kronheimer's book on the geometry of four manifolds, a brief review of connections on principal bundles is given. Three equivalent definition are stated: 1) Via horizontal subspaces, ...
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0answers
49 views

If a thread is pulled out of a floating blob of water, must the thread be tangent to the surface of the blob at some point?

My motivation is the recent question I just answered, and my answer use too many hypothesis that I considered superfluous: Always "one double root" between "no root" and "at ...
0
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0answers
19 views

Show differential of $f:S^{m}\times S^{n}\to S^{m+n+mn}$ is injective

The problem is find to the differential of $f:S^{m}\times S^{n}\to S^{m+n+mn}$ (spheres) defined as $f(x_{0},...,x_{n},y_{1},...y_{n})=(x_{0}y_{0},x_{0}y_{1},...,x_{n}y_{1},x_{n}y_{n})$ and show it is ...
0
votes
1answer
24 views

Differential of rotation matrix at the north pole of sphere

Let T(p) rotate $p\in S^{2}$ by angle $\theta $ about the z-axis. The problem is to compute $dT_{(0,0,1)}$. T can be represented by the usual 3x3 rotation matrix $A_{z}(\theta)$. So $T(p)=A_{z}p$. ...
0
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0answers
28 views

Coordinate frames along the bounday of a minimal area (soap-film) surface

I would like to calculate coordinate frames along a closed Bezier (Or Catmull-Rom) spline. One axis should be tangential to the curve, and another axis normal to the minimal-area surface (soap-film ...
1
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1answer
29 views

Difference betwee parameterization and embedding of manifolds

What is the difference between embedding and parameterization? Why, for example, we say Gauss parameterization of a convex hypersurfaces, and we don't call it an embedding?
7
votes
2answers
113 views

Smooth surfaces that isn't the zero-set of $f(x,y,z)$

The zero-set of any smooth function $f(x,y,z)$ with a non-vanishing gradient is a smooth surface. I was wondering if the reverse is true: is every smooth surface in $E^3$ the zero-set of some smooth ...
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3answers
63 views

May directed graph be embedded into manifold?

May directed graph be embedded into manifold?How ?and what is the condition?
2
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2answers
35 views

Calculus III: Find the points of the curve…

I have to find the points of the curve $$r\left( t \right) =\left( t,{ t }^{ 2 },{ t }^{ 3 } \right) $$ where the osculating plane passes through the point $\left( 2,-\frac { 1 }{ 3 } ,-6 \right)$.
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The name of this metric.

Does anyone know what metric is the following one in $\mathbb{R}^n$? $g=\sum_{i=1}^{n}(1+x_i^2)dx_i^2+2\sum_{i\neq j}x_i x_j dx_i dx_j$ Thanks in advance.
2
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0answers
18 views

Reference for Envelope, Evolute and involute

I have to give a lecture on Envelopes, Evolute and Involute to I year undergraduate students. Please suggest me some books which explain these concepts with examples geometrically. Already I have seen ...
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1answer
15 views

contraints on equation of a cylinder

(x-a)^2+(y-b)^2 = r^2 any way to adjust this formula to add constraints to the z-axis? recently introduced to the idea this goes forever in z-axis and I want to see if theres way's to adjust formula ...
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Zeros of vectorial field [closed]

Given a $M$ manifold in ${\mathbb R}^n$ and $X:M\rightarrow TM$ a vectorial field such that $\pi\circ X=Id$ where $\pi:TM\rightarrow M$ (projection to $M$). One zero of $X$ is such that ...
6
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1answer
38 views

Flat connection with non-trivial holonomy? I cannot get it

maybe this is a dumb question, but I cannot understand how a principal $G$-bundle can have non-trivial holonomy with a flat connection. Maybe I'm missing something, but doesn't Ambrose-Singer theorem ...
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1answer
29 views

A bounded domain can be considered as a compact manifold?

A bounded domain $\Omega$ with smooth boundary $\Gamma$ can be considered as a compact connect Riemannian manifold?
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2answers
34 views

Is it possible to write the curl in terms of the infinitesimal rotation tensor?

Is it possible to write the curl in terms of the infinitesimal rotation tensor? Basically, we can write the curl as a matrix operator $$ curl=\begin{bmatrix} 0 & -\partial z & \partial ...
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0answers
24 views

Easy solution to Yamabe problem for surfaces

The Yamabe problem asks if, given a Riemannian manifold $(M,g_0)$, it is possible to find a conformal metric $g$ on $M$ with constant scalar curvature. I would like to know if there is some "easy" ...
0
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1answer
30 views

calculate normal field of cylinder and sphere

in a book it is given that the unit normal field of $S^2$ is $N(p)=p$ the identity map. pictorially it is clear to me. But if I take any point on the sphere and multiply(usual scalar product of ...