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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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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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
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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
17 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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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
38 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
34 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
24 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
39 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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0answers
20 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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42 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
34 views

Does the decomposition of a Lie group manifold imply a type of group product?

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
41 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
65 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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0answers
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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23 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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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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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 $ ...
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1answer
44 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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1answer
99 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
34 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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24 views

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
43 views

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
21 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
67 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
23 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
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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
35 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 ...
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1answer
40 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
28 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 ...
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2answers
96 views

If the curvature of a curve in a plane is a constant, is the curve contained in a circle?

If the curvature of a curve in a plane is a constant, is the curve contained in a circle?(Suppose curvature is positive.) one of my homework problems needs to use this, but I am not sure whether this ...
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0answers
23 views

Recover a surface's equation from its curvature

Can the equation of a surface in Euclidean 3 space be recovered from the equation of its Gaussian curvature?
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1answer
51 views

Closed non-exact $2$-form on $T = S^1 \times S^1$

I would like to find a closed non-exact $2$-form on $T = S^1 \times S^1$. Here are my thoughts: Since $d\theta$ and $d\varphi$ are closed non-exact $1$-forms an obvious candidate is $d\theta \wedge ...
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1answer
30 views

Finding two inequivalent closed, non-exact $1$-forms on $T = S^1 \times S^1$: second check

This is a follow up on my previous question. I would like to test whether I understand the first part of the answer given to me there by rewriting it in my own words. Please could someone tell me ...
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1answer
54 views

What class would this be covered in?

Would the material in this section of a wikipedia article be covered in a standard course on Differential Geometry, or should I look elsewhere to learn those sorts of things? Specifically, topics like ...
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1answer
61 views
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Gluing submanifolds along their common boundary

Let $M$ be a smooth $m$-manifold and $N_1$, $N_2$ smooth embedded $k$-submanifolds such that $N_1\cap N_2=\partial N_1=\partial N_2$, for each $x\in N_1\cap N_2$, $T_x N_1=T_x N_2$, and for $x\in ...
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1answer
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the top chern class of the holomorphic tangent bundle is the euler class

Is the following true? Let X be a complex manifold of complex dimension d and let V denote its holomorphic tangent bundle (ie it's $T^{1,0} \subset T \otimes_R C$, where T is the tangent bundle of ...
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48 views

Exterior Algebra: Characterization [on hold]

Just a short question: Given a vector space. Why is the exterior algebra characterized as: The largest anticommutative integer-graded algebra with identity linearly embedding the vector ...
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2answers
48 views

A doubly ruled surface which is not a plane must be quadratic

I want to show that a doubly ruled surface which is not plane must be quadratic. Any help will be appreciated.
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1answer
62 views

Pullback of a differential form

My question is in regards to a proof in Lee's 'Introduction to Smooth Manifolds'. He proves a lemma about the pullback of a differential form on a manifold $N$, where $F:M\rightarrow N$ is a smooth ...
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2answers
51 views

Prove that $\frac{dN}{ds}=-\kappa T$

Prove that $\frac{dN}{ds}=-\kappa T$, where $N$ is the oriented normal, and $T$ is the unit tangent vector, and $s$ is arc-length parameter. Here's what I've got so far from my note and I don't ...
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3answers
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Embeddings are precisely proper injective immersions.

We call a map $f: X \to Y$ between topological spaces proper if $f^{-1}(K)$ is compact for all compact $K \subset Y$. Where can I find a reference that embeddings are precisely proper injective ...
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Question concerning tensors

As some of you may have seen from my previous question(s), I am working through Spivak's Calc on Manifolds and this happens to be the first time I've been introduced to tensors formally, though I have ...
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1answer
72 views

Finding two inequivalent closed, non-exact $1$-forms on $T = S^1 \times S^1$

I've been studying the torus and the first cohomology group $H^1_{dR}(T)$ for a couple of weeks now. I finally had a breakthrough of understanding and would like to kindly request the community to ...