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Questions tagged [differential-geometry]

Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses on "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such structures, use (differential-topology) instead. Use (symplectic-geometry), (riemannian-geometry), (complex-geometry), or (lie-groups) when more appropriate.

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Isometry between superfaces

Let $S$ be the superface $S = \{(x,y,z) : x^2 + y^2/4 = 1\}. $ Minding theorem says that $S $ and $C $ the cylinder $x^2 + y^2 = 1$ are locally isometric, but can anyone build an isometry between $S ...
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1answer
49 views

Soft question: why define smooth manifolds intrinsically?

A possibly naive question, but one I've been grappling with since starting Riemannian geometry. I have done some searching and I cannot find this question asked, yet it feels like a very natural one ...
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0answers
10 views

Curvature and torsion of spherical curves

I am trying to understand the properties of curves (especially closed curves) that live on the surface of a sphere. What are the implications on the curvature and torsion of a curve, of living at a ...
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1answer
21 views

Curvature and torsion of circular helix

$c:r=r(s)$ $ c$ is circular helix iff $ det(r’’,r’’’,r^{(4)})= 0 $ i will prove one side but other side i said : Let $det(r’’,r’’’,r^{(4)}) = 0 $ then , $r^{(4)} = \alpha(s) r’’ + \beta(s) r’’...
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0answers
30 views

Holomorphic form an a blow up

Consider the blow up of $\mathbb C^2/\mathbb Z_2$ at its singularity $0$. Since $dz_1\wedge dz_2$ is invariant under $z\mapsto -z$, it passes to a well defined holomorphic form on $(\mathbb C^2/\...
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19 views

Gauss formula along a curve

On page 138 of "Riemannian Manifold: An introduction to curvature" I can read the following statement: Let $S$ be a Riemannian submanifold of $M􏰄$, and $\gamma$ a curve in S. For any vector field $...
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0answers
12 views

Principal Symbol for Ricci-DeTurck Flow

I am following some lecture notes on Ricci flow and linearizing an operator to obtain its principal symbol. We have $T \in \: \Gamma(Sym^2 T^{*}M)$ smooth, fixed and positive definite and then ...
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Induced Maps from Sections

I am reading a book on Riemannian geometry and it says 'let $T \in \:\Gamma(Sym^2 T^{*}M)$ be fixed, smooth, and positive definite. We also denote by $T$ the invertible map which $T$ induces, given a ...
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1answer
27 views

A different structure in S1?

I want to know another structure in S1. I want that it not be diffeomorphic to the usual structure. Thanks!
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1answer
33 views

Visualizing tangent space from its definition

Let me quote Guillemin, Pollack here. "We can use derivatives to identify the linear space that best approximates a manifold $X$ at a point $x$. Suppose $X\subset R^n, \phi:U\rightarrow X$ is a local ...
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1answer
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$L^p$ space of vector fields is complete

Let $(M,g,\mu)$ be a weighted Riemannian manifold. I want to show that the $\vec{L}^p$ space of measurable vector fields on $M$ is complete. I tried using the fact completeness is equivalent to ...
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3answers
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Tangent vector field to a smooth curve over a smooth manifold

I am teaching myself some elementary differential geometry and am stuck on the concept of the tangent vector field of a smooth curve. I have searched the web for an hour or so but cannot find anything ...
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1answer
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$f,g:U^2\to\Bbb R$ smooth. Is $\bar h(x,y)=\int_Uf(x,z)g(z,y)dz$ smooth?

$f,g:U^2\to\Bbb R$ smooth. Is $\bar h(x,y)=\int_Uf(x,z)g(z,y)dz$ smooth? Under the convention that smooth means element of $C^\infty$, I know that could be rewritten as $\bar h(x,y)=\int_U h_{x,y}(z)...
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1answer
18 views

Definition of closed surface/manifold

This question might appear silly, I was reading on wikipedia that a closed surface (or manifold in general) is a surface without a boundary, I'd like to elaborate a bit on such definition. Assuming ...
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0answers
19 views

Flux of a vector field in $S^n$

I need compute the flux of $X = \sum_{i =1}^n x_i \partial_{x_i} $ in $S^n $ with the stereographic proyection $\Pi (x_1,...,x_{n+1}) = \left(\frac{x_1}{1 - x_{n+1}},...,\frac{x_n}{1-x_{n+1}}\...
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0answers
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Need help with the proof of Ricci tensor of Einstein and anti self dual manifold.

Let M be an anti-self dual and Einstein 4-manifold with scalar curvature s. Then the Ricci tensor $c_Z$ of the twistor space is given by $$c_Z (E,E) = (s/4 - t(s/12)^2) \|X\|^2 + (1+ (ts/12)^2)\|V\|^...
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0answers
26 views

Normal vector at boundary of a surface

Given an orientable, co-dimension 1 surface $\mathbb{S} \in \mathbb{R}^3$ with boundary $\partial \mathbb{S}$, is there a commonly-accepted definition of the outward normal vector to the surface at $\...
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1answer
33 views

Codifferential / divergence of differential form under conformal metric change

I have a question related to this and a second post. I want to calculate the codifferential under a conformal metric change, $g_\psi = e^{2\psi} g$. By Besse's book on Einstein manifolds, or an ...
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1answer
30 views

Prove that the covariant derivative commutes with musical isomorphisms

Suppose I have a covector field $\omega$ and a covariant derivative $\nabla_{X}$ for some vector field $X$ on a Riemannian manifold $(M, g)$. Define $X^{\flat} \in \mathfrak{X}^{*}(M)$ as $X^{\flat}(...
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1answer
30 views

Tangent spaces via derivations.

I am a little stumped by the definition of the tanget space via derivations. https://en.wikipedia.org/wiki/Tangent_space#Definition_via_derivations Is it reasonable to think of these as "directional ...
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0answers
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Smooth functions and equalizers

Preliminaries: Let $(\mathcal{G}_0,\mathcal{G}_1)$ be a Lie groupoid. In particular, $\mathcal{G}_0,\mathcal{G}_1$ are smooth manifolds and we have smooth maps $s,t:\mathcal{G}_1\to \mathcal{G}_0$ ...
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1answer
22 views

Proving these spaces are not surfaces

Hi all, Here I have two problems that really I can´t know how solve. For $X = S^2 \cup \{x_0\}$ where $x_0 \in \mathbb{R}^3 \setminus S^2$, being $S^2$ the sphere. In $X$ we consider the usual ...
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1answer
42 views

Spaces of submanifolds

Let $M$ and $N$ be smooth manifolds with $\dim M<\dim N$. The spaces $\mathrm{Emb}(M,N)\subset\mathrm{Imm}(M,N)$ of smooth embeddings and immersions $f:M\to N$, respectively, are infinite ...
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0answers
29 views

Does a surface $S$ diffeomorphic to a sphere have an umbilical point?

I should see if a surface which is diffeomorphic to a sphere in $\mathbb{R}^3$ always has an umbilical point. So, I start by locally parametrizing the sphere with a diffeomorphism $x: \Omega \to S^2$ ...
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1answer
25 views

Isometries of a surface with metric of curvature $-1$

Let $\mathbb{H}^2$ be the hyperbolic space in the model $$\mathbb{H}^2=(\mathbb{R}\times\mathbb{R}_+,g=\frac{1}{y^2}(dx^2+dy^2)).$$ It is known that the Mobius transformations, with $ad-bc=1$, are ...
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0answers
26 views

Killing vectors corresponding to the Lorentz transformations

I have a problem with one thing. Let's consider the Lorentz group and the vicinity of the unit matrix. For each $\hat{L}$ from such vicinity one can prove that there exists only one matrix $\hat{\...
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0answers
47 views

Resolution of singularity of $\mathbb C^2/{\mathbb Z_2}$ (blow up)

Consider the $\mathbb Z_2$-action $g:\mathbb C^2\to \mathbb C^2, z\mapsto-z$ on $\mathbb C^2$ and its quotient $X:=\mathbb C^2/{\mathbb Z_2}$. This is a singular surface with singular point the image ...
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0answers
10 views

Enneper surface is an injective inmmersed surface

Show that the map $\varphi:\mathbb{R}^2\to \mathbb{R}^3$ defined by, $$\varphi(u,v)=\left(u-\frac{u^3}{3}+uv^2,v-\frac{v^3}{3}+vu^2,u^2-v^2\right),$$ is an injective immersed surface. The problem is ...
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0answers
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Second derivate: coordinate free version is equivalent to normal version.

Let $f: A \to B$ be a differentiable function. I know that $Df: A \to Hom(A,B)$ and that $(D^2f): A \to Hom(A,Hom(A,B)) \cong L(A,A; B)$. Furthermore, my teacher proved in class that if $f \in C^2$ ...
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0answers
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Change of coordinates and effects on the tangent bundle of a Manifold

I am doing ex. 2.3(2) from Frankel's book "The geometry of Physics". He says to consider the tangent bundle to a manifold M and show: i) that under a change of coordinated in M, $\partial / \partial ...
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1answer
38 views

how we can add a 2-tensor to a 1-form?

In the definition of a Randere norm, we add a Riemannian metric $\alpha$ to a 1-from $\beta$. Indeed, $F(y)=\sqrt{a_{ij}(x)y^iy^j}+b_i(x)y^i$ in ehich $\alpha(y)=\sqrt{a_{ij}(x)y^iy^j}$ is a ...
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0answers
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Lie derivative and Weyl Tensor

Let's consider a conformal symmetry such as : $\mathcal{L}_X \ g_{\alpha \beta}=2 \psi(x)g_{\alpha \beta}.$ How to prove that ${\mathcal{L}}_X \ W_{\alpha \beta \gamma \delta}=0$, where $W$ is the ...
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1answer
25 views

What is the natural isomorphism between the tangent space of a product manifold and the product of the tangent spaces?

Let $M$ and $N$ be smooth manifolds and $p\in M$, $q \in N$. What is then a natural isomorphism for $$T_{(p,q)}(M\times N) \cong T_pM \times T_pN ?$$
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Show that each point of closed submanifold $\mathbb R^{n+2}$ have “tangent line” in $\mathbb R^{n+2}$ [duplicate]

Let $n \geq 1$ be an integer and $M \subset \mathbb { R } ^ { n + 2 }$ a smooth $n -$ dimensional submanifold which is a closed subset of $\mathbb { R } ^ { n + 2 } .$ Prove that for any $x _ { 0 } \...
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0answers
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In differential geometry, to what extent does the curvature tensor determine its associated connection?

In the absence of a metric, it is not clear to me to what extent does knowing the curvature tensor determine its associated connection? I would be satisfied knowing this for zero torsion. I'd like to ...
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1answer
15 views

Search singular point of a map on ball to Euclidean Space

I know the form of map $f$ told us the singular point of this map would be symmetry, but I still do not know which conditions I should to use the to solve the problem or what techniques I need for ...
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1answer
25 views

Differentiable curve of matrices such that the derivative at 0 has trace equal to zero.

Suppose that $\epsilon > 0$, $\gamma: (-\epsilon,\epsilon) \to \text{End}(\mathbb{R}^n)$ is a differentiable curve of matrices such that $\text{det}(\gamma(t)) = 1$ for all $t\in(-\epsilon,\epsilon)...
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0answers
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Dominance of two real-valued funstions on manifold

I have two $\mathbb{R}$-valued functions $f,g : G \to \mathbb{R}$ on a Lie group $G$. For example, $G = SO(3)$. The two functions $f,g$ are ugly and it is hard to compute the exact value of $f(M), g(M)...
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1answer
32 views

Differential geometry project ideas for undergraduates [on hold]

I’m a 4th year physics undergraduate student and I just took an introductory differential geometry course (started from the basics of a manifold to the basics of different kinds of curvatures in a ...
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1answer
18 views

interaction of left-invariant vector fields and right-translation on a Lie-Group

Given a Lie-Group $G$ denote the set of left-invariant vector fields on $G$ by $LG$ and denote by $R_g$ the right-translation, i. e. for $g \in G$ define $$R_g \colon C^\infty (G) \to C^\infty (M) \...
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2answers
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How do I show $X_{\omega(Y,Z)}=-[Y,Z]$?

How do I show $X_{\omega(Y,Z)}=-[Y,Z]$, where $\omega$ is a symplectic 2 form (in particular non-degenerate) and $Y,Z$ are vector fields and $X_f$ is the vector field correspond to the 1 form $df$ ...
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1answer
38 views

Self intersection and cohomology of boundary of tubular neighbourhood

Let $X$ a compact orientable manifold of dimension $2n$ and $Y$ a compact submanifold of dimension $n$. Further let $U$ a tubular neighbourhood of $Y$. When I did some calculations I got the ...
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0answers
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Search singular point of map on ball in Euclidean Space [on hold]

I know the form of map f told us the singular point of this map would be symmetry, but I still do not know which conditions I should to use the to solve the problem or what techniques I need for this ...
0
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1answer
39 views

Diffeomorphism Preserves Tangency (Do Carmo 2.4.25)

Suppose $C_1$ and $C_2$ are regular curves on a regular surface $S$. Suppose $p$ is a point in $S$ where $C_1$ and $C_2$ are tangent, then if $\varphi:S\rightarrow S$ is a diffeomorphism, prove that $\...
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0answers
27 views

Define a parameterization given a vector field on a smooth manifold

I'm not aware of any techniques or theorems that would give an answer to the following question. Suppose we have a smooth surface $\mathcal{S} \subset \mathbb{R}^3$, and also suppose that for each ${...
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1answer
23 views

Question Regarding Relatively Open Sets of a Submanifold

I am reading a text in introductory differential geometry and it says " If $M$ is a submanifold of $\mathbb{R}^n$, and $N$ is open relative to $M$, then it follows easily from the definitions (of a ...
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0answers
33 views

Topology on the affine space of connections

What is the natural topology we generally define on the Affine space of Connections? I am not able to find any literature where this topology is explicitly described. It would be really ...
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1answer
23 views

Chapter 2.4 Exercise 1 in Do Carmo (Tangent Plane)

Show that the equation of the tangent plane at $p=(x_0,y_0,z_0)$ of a regular surface is given by $f(x,y,z)=0$ where $0$ is a regular value of $f$, is $$f_x(p)(x-x_0)+f_y(p)(y-y_0)+f_z(p)(z-z_0)$$
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1answer
27 views

Prove the lines from the foci to a point on an ellipse form equal angles with any tangent vector at that point

Suppose we have an ellipse $\frac{x^2}{p^2}+\frac{y^2}{q^2}=1$ with parametrization $\gamma(t)=(p\cos(t),q\sin(t))$. Let $\vec{p}= \gamma(t_0)=(p\cos(t_0),q\sin(t_0))$ be any point on the ellipse. Let ...
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0answers
15 views

Tangent spaces of two transverse subspaces are transverse subspaces

I am very new to differential geometry and was thrown this very long question: Suppose that two subspaces $V$ and $W$ of $\mathbb{R}^n$ are transverse (so $\text{Span}(V,W)=\mathbb{R}^n$). Let $O$ be ...