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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Why is Klein bottle non-orientable?

I am doing the homework of differential geometry and encounter this problem: The Klein bottle $K^2$ is defined to be the identification space $$[0, 1] \times [0, 1]/{\sim}, \text{ where the ...
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12 views

Help visualizing Ricci scalar curvature?

I am trying to learn more about Ricci scalar curvature. I am trying to get an image in my head of what scalar curvature actually represents about the curvature of a manifold. The most familiar image I ...
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1answer
22 views

What's the difference between a directional derivative and a derivation?

I asked my uncle what a derivation is and and he wrote the following: Most calculus courses discuss directional derivatives and include geometric applications to surfaces of the form ...
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0answers
23 views

Is every $S^1$ knot orientable?

let $K$ be an $S^1$ knot, i.e., $K$ is a homeomorphic copy of the "standard" $S^1$ living in $\mathbb R^3$ ). I am trying to see if $K$ is orientable. This is what I have: 1) If the map sending $S^1$ ...
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14 views

An integral with respect to the Haar measure on a unitary group

Let $A,D\in \mathbb{C}^{n \times n}$ be diagonal matrices. I need to calculate $$\int_{U(n)}\det{(A-HDH^\dagger)}\,\mathrm{d}H$$ where $dH$ is the unit invariant Haar measure on the group of unitary ...
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0answers
16 views

Flat connection of a vector bundle over a 1 dim. manifold

I'd like to show that a connection of a vector bundle $E$ over a 1 dim. manifold $M$ is flat, or equiv. that its curvature is zero. Let $D$ denote the connection, $\sigma$ a section of $E$ and $v,w$ ...
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1answer
12 views

Projection map from $\mathbb R^n-(0)$ to $\mathbb RP^{n-1}$ is smooth

How do I show that the projection map from $\mathbb R^n-(0)$ to $\mathbb RP^{n-1}$ taking $x$ to its equivalence class $[x]$ is smooth?
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16 views

Prove the following statement: [on hold]

If $\hat \zeta \epsilon se(3)$, show that $g\hat \zeta g^{-1}\epsilon se(3)$ where $g\epsilon SE(3)$
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17 views

Tangent space of a Product of two manifolds

Suppose $M$ and $N$ are two $C^\infty$ manifolds. Take $p\in M$ and $q\in N$. We have the following maps between these: $\iota_1 : M\to M\times N$, $\iota_2:N\to M\times N$, $\pi_1:M\times N\to M$ and ...
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12 views

References for mean curvature flow in Riemannian manifolds

I am interested in mean curvature flows (MCFs) in Riemannian manifolds. But most textbooks about MCF seem to treat mainly MCFs of hypersurfaces in $\mathbb{R}^{n+1}$. Should I study them before ...
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9 views

How to find the points of self intersection of Cayley's Sextic?

I am given that $Y(t)=\cos^3(t)(\cos(3t),\sin(3t))$ and need to find the unique point of self intersection. So I assumed $$\cos^3(t)(\cos(3t),\sin(3t))= \cos^3(u)(\cos(3u),\sin(3u)).$$ I took lengths ...
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21 views

To Prove that The Level Set Of AConstant Rank Map is a Manifold

Let $f:\mathbf R^n\to\mathbf R^m$ be a smooth function of constant rank $r$. Let $\mathbf a\in \mathbf R^n$ be such that $f(\mathbf a)=\mathbf 0$. Then $f^{-1}(\mathbf 0)$ is a manifold of ...
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1answer
57 views

Why do we require that a complex manifold has the structure of a real manifold?

I am taking a course in complex manifolds, heavily influenced by Huybrechts' book "Complex Geometry", and in it we define a complex manifold $X$ to be a smooth, real manifold $M$ together with an ...
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7 views

Dynamical systems,forward invariant [on hold]

Show that the complement of a forward invariant set is backward invariant, and vice versa. Show that if f is bijective, then an invariant set A satisfies f t (A)= A for all t. Show that this is false, ...
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3answers
41 views

Does parallel transporting require an ambient space?

Can someone summarize why an ambient space isn't needed to measure curvature when parallel transporting tangent vectors or vector fields along a curve on a Riemannian manifold? How do we define the ...
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23 views

Determine the equation of the plane . [on hold]

We have given: $$ 1) \ y= x \\ 2) \ y = x^2 $$ Now, I need to determine the equation of the plane for 1) and 2). Help. Thanks in advance.
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2answers
17 views

What's the name of a solid that results from extruding an area straight along an axis?

If you have any kind of 2D shape and move it up into the third dimension, what do you call it, because it seems like prism is used only if the base is a polygon. It also seems like extrusion is a ...
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0answers
20 views

a question about differential geometry, the relation between osculating plane and the points of $\alpha(s)$

question:please prove the limit position of the circle passing through $\alpha(s)$,$\alpha(s+h_{1})$,$\alpha(s+h_{2})$ when $h_{1}$ and $h_{2}$ approaches 0 is a circle in the osculating plane at s, ...
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2answers
43 views

If a curve $\gamma$ through two points $P,Q$ satisfy $\|Q-P\| = \int^{t_1}_{t_0} \| \gamma^{'} \| \, \text {d}t$, then $\gamma$ is a straight line?

In a theorem called "A straight line is the shortest curve through two given points", I prove that for any two points $P,Q \in \mathbb R^2$ and any curve $\gamma : (a,b) \rightarrow \mathbb R^2$ with ...
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0answers
42 views

What is this inner product on differential forms?

I am trying to understand the definition of $d^\ast$ of $d$ where $d$ denotes the exterior derivative as given in these lecture notes. (please see page 3) Here are my thoughts so far: Let us ...
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1answer
37 views

Is it known whether $S^6$ is a Kähler manifold?

I have just started to learn about Kähler manifolds and I now am wondering: Is it known whether $S^6$ is a Kähler manifold? By definition a Kähler manifold has 3 structures: a symplectic, a ...
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2answers
26 views

Integrating the Riemannian volume form

Let $M$ be a compact manifold with $\partial M = \varnothing$ and let $\omega$ be the volume form $\sqrt{\det g_{ij}} dx_1 \wedge \dots \wedge dx_n$. I want to show that $\omega$ is not exact. My ...
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2answers
40 views

Some question about this proof about Riemannian volume form

In these lecture notes lemma 2.3. is given as $\omega_g = \sqrt{\det g_{ij}} \, dx^1 \wedge \cdots \wedge dx^n$ is independent of the choice of coordinate charts. I am trying to understand the ...
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21 views

Sectional curvature in 3-dimensions

I wonder how to compute the sectional curvature of 3-dimensional objects eg. unit ball, $H=\{(x_{0},x_{1},x_{2},x_{3})\in \mathbb{R}^{4}:x_{0}^{2}-(x_{1}^{2}+x_{2}^{2}+x_{3}^{2})=1$ and ...
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28 views

Does there exists known special cases of a zero Riemann tensor for 3D metrics?

In two dimensions, if one has a flat metric $g_{ab}$, then one can transform $g_{ab}$ to another flat metric $h_{ab}=e^{2\varphi}g_{ab}$, when $\nabla^2 \varphi =0$ and the Riemann tensor remains ...
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49 views

one question about differential geometry,show the curvature k($\phi$)

One often gives a plane curve in polar coordinates by $p=p(\phi)$,$a\le\phi \le b$. (1)Show that the arc length is $$\int_{a}^{b}\sqrt{p^2+\dot p^2}$$,where $\dot p$ means the derivative of p with ...
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0answers
46 views

Decomposition of a group manifold; is there an associated group decomposition?

The real symplectic group manifold is diffeomorphic to this Cartesian product of manifolds: \begin{equation} \operatorname{Sp}(2n,\mathbb{R}) \simeq \operatorname{U}(n) \times \mathbb{R}^{n(n+1)}. ...
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1answer
43 views

Matrix representation of shape operator

Let $f$ be a parametrized surface $f: \Omega \subset \mathbb{R}^2 \rightarrow \mathbb{R}^3$ and $N : \Omega \rightarrow Tf$ the Gauß map. Then the shape operator is defined as $L = -DN \circ Df^{-1}.$ ...
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1answer
45 views

How to smoothly extend a function?

Here is what I am trying to do: Let $X$ be a paracompact smooth manifold. Let $C$ be closed, $U$ open and $C\subset U \subset X$ and $f$ is a smooth map on $U$. I want to show that then there ...
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1answer
66 views

What is the step in this proof “because $\omega$ is closed”?

I am working through this proof of the Poincare lemma here but I don't understand one step. First, there is the following equation $$ {\partial \over \partial x^j} f(x) = \int_0^1 \left (t ...
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1answer
32 views

How to prove that volume forms agree on $U_\alpha \cap U_\beta$?

I am familiarizing myself with Riemannian manifolds. Let $M$ be an orientable smooth $n$-manifold with atlas $(U_\alpha, (x_1^\alpha, \dots, x_n^\alpha))$ and let $g$ be a Riemannian metric on $M$. ...
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74 views

Show $f:\mathbb{R}^n \to \mathbb{R}^m$, $n>m$ can't be 1-1

Problem 2-37 on p. 39 of Spivak's Calculus on Manifolds asks Let $f:\mathbb{R}^2 \to \mathbb{R}$ be a continuously differentiable function. Show that $f$ is not 1-1. (Hint: If, for example, ...
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1answer
36 views

When is a curve parametrizable?

Is there a way in general to tell whether a given curve is parametrizable?
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24 views

Tangential derivative vs covariant derivative

My question is basically the same as this, but the answer in that page was not clear to me. Let me restate the question here: let $\Omega\subset\mathbb{R}^3$ be a domain with boundary $\Gamma$, and ...
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2answers
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Notation of coordinate representation in Lee

In Lee's Introduction to Smooth Manifolds he writes $$ \omega = \omega_i dx^i$$ where $\omega$ is a differential form. See for example page 293. What does $\omega_i dx^i$ stand for? According ...
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52 views

Is my understanding of the argument correct?

I worked through a proof of: $$ f(z) = {1\over 2 \pi i}\int_{\partial D} {f(w) \over w -z} dw$$ where $D\subset \mathbb C$ is an open disk and $f$ is holomorphic on $D$ and continuous on ...
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0answers
31 views

Name of isometric invariant in Gauss-Bonnet

Does the tangential rotation term $ \int k_g ds $ of Gauss-Bonnet theorem ( for continuous or discontinuous lines on a surface) have a name or symbol in differential geometry ? The second term $ ...
2
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1answer
45 views

Can I argue like this to prove that the determinant is positive?

Let $X$ be a smooth $n$-manifold with an oriented atlas $\mathcal U = (U_\alpha, (x_1^\alpha, \dots, x_n^\alpha))$. Let $g$ be a Riemannian metric on $X$. Let $g_{ij} = g\left ( {\partial \over ...
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2answers
116 views
+100

Proof in Hamilton: Divergence theorem for differential forms?

For a vector field $X\in\Gamma(TM)$ on a closed Riemannian manifold $(M,g)$ we have \begin{align*} \int_M\text{div}X\;\mu=0, \end{align*} where \begin{align*} ...
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1answer
25 views

Flows: Escape Lemma

Denote compact sets by: $\mathcal{C}$ Given a smooth manifold. Consider the maximal flow of a smooth vector field: ...
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1answer
35 views

Simple Differential Geometry/Analysis question: Prove that $f:\mathbb{R^2}\to\mathbb{R}$ is continuous

In Differential geometry of curves and surfaces by Manfredo do Carmo, page 459, it says the following: Observe that $f:\mathbb{R^2}\to\mathbb{R}$ is continuous, where $f(x,y) = \frac{x^2}{a^2} - ...
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Parametrization of surfaces gauss

What is pde relating $f$ and $g$ if the Gauss curvature is $+1$ and $-1$ respectively: $$ \begin{align*} (x,y,z) &= ( u \cos(v), u \sin(v), f(u,v) ) \\ (x,y,z) &= ( u \cos(v), u \sin(v), ...
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1answer
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What is the anticommutator of the interior product and codifferential (adjoint of exterior derivative)?

What is $\eta=i_X \delta + \delta i_X$ acting on differential forms? Here $\delta$ is the usual Hodge adjoint of the exterior derivative and $i_X$ is contraction of a form with the vector field $X$. ...
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1answer
23 views

Stereographic projection of a sphere

What should have been a simple exercise in geometry has morphed into a multi-day affair with me figuratively tearing my hair out. I have no clue what's wrong. This image accompanies the problem: ...
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1answer
68 views

Cauchy integral formula: can it be proved like this?

Consider the Cauchy theorem: Let $D\subset \mathbb C$ be a domain such that $\partial D$ is smooth and $\overline{D}$ is compact. Let $f$ be holomorphic on $D$ and continuous on the closure. Then ...
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1answer
28 views

Paramatrizes curve with constant speed

Show that if $\alpha : I \rightarrow \Re^{n+1}$ is a parametrised curve with constant speed then $\alpha(t) \perp \frac{d}{dt} \alpha(t)$ for all $t\epsilon I$.
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1answer
51 views

First encounters with sheaves: could you tell me if my thoughts are correct?

Let $X=\mathbb R$ be the reals and $\mathbb Z$ the integers (with the discrete topology). Do I understand correctly the definition of sheaf? Here is how I understand it (illustrated by a ...
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2answers
36 views

Why are derivations useful for defining tangent vectors?

On page 54 in his book Introduction to Smooth Manifolds, John Lee says the following: A linear map $v: C^\infty (M) \rightarrow \Bbb{R}$ is called a derivation at p if it satisfies ...
3
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1answer
43 views

Map between Tangent Manifolds Well-Defined?

Let $f: \mathcal{M} \to \mathcal{N}$ be a $\mathscr{C}^{r+1}$ map. We define a map $\mathscr{T}f: \mathscr{T}\mathcal{M} \to \mathscr{T}\mathcal{N}$ as follows: A local representation of the map ...
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1answer
29 views

critical point for the curvature does not correspond to a local maximum/minimum.

Draw an example where a critical point for the curvature does not correspond to a local maximum/minimum Does the curve for infinity sign satisfy this? I am having trouble seeing why it's true, if of ...