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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Morphism of vector bundles covering maps of the bases

Let $E_{1}\to B_{1}$ and $E_{2}\to B_{2}$ be vector bundles over differentiable manifolds $B_{1}$ and $B_{2}$. What means that $F\colon E_{1}\to E_{2}$ is a morphism of bundles covering a map $f\colon ...
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264 views

Stokes theorem for manifolds with corners

I wonder if you could recommend a chapter or a paper on Stokes theorem for manifolds with corners. I've found one here http://math.stanford.edu/~conrad/diffgeomPage/handouts.html (the third one from ...
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75 views

Finding a local parameterization of a plane curve

I'm attempting to find a parameterization of $\frac{x_1^2}{a^2} + \frac{x_2^2}{b^2} = 1$. I find a tangent vector field: $X = \left( \frac{2x_2}{b^2}, -\frac{2x_1}{a^2} \right)$ (by taking the ...
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Lie group. How is manifold defined?

If I have the Lie group $SL(2,\mathbb{R})$. Then how is the manifold structure on this algebraic group defined, could anybody explain this to me? I mean this is the group of matrices that determinant ...
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Construct two-form

I should give an example and construct a two-form on the 2D sphere. I know how to construct one-form on the 2D sphere, but I have no idea how to continue with the two-form.
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A question on the curvature of a regular parametrized curve

The next question is from Do-Carmo's baby book, page 30 question 3 in section 1-6. The question goes as follows: Show that the curvature $k(t)\neq 0$ of a regular parametrized curve $\alpha : ...
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149 views

What is the canonical bundle of a smooth divisor?

I am currently trying to learn some complex geometry, using mainly the book by Huybrechts. There is one thing confusing me, however: For example on Wikipedia people are talking about the canonical ...
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22 views

Show that if $[III]=\lambda[I]$ for some function $\lambda$ on the surface, then either $K=0$, or the surface is part of sphere.

Show that if $[III]=\lambda[I]$ for some function $\lambda$ on the surface, then either $K=0$, or the surface is part of sphere. Here's what I got: We know that $[III]-2H[II]+K[I]=0,$ so ...
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2answers
80 views

Proving that $\Delta(M \times M)$ is a submanifold of $M \times M$

I am struggling to prove that $\Delta(M \times M) = \{(x,x) : x \in M\}$ is a submanifold of $M \times M$. A manifold M is a submanifold of N if there is an inclusion map $i:M \rightarrow N$ ...
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81 views

What is the exterior algebra?

I am learning differential geometry, and I have difficulty understanding the construction of the exterior algebra of an $n$-dimensional vector space $V$. We have the wedge product ...
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The Operator '$d$' Apparently Having two Different Meanings in Differential Geometry.

Given a smooth map $f:M\to N$ between smooth manifolds $M$ and $N$, we denote the global differential of $f$ by $df$. Also, the letter '$d$' is used for denoting exterior derivative of a differential ...
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48 views

What exactly is meant by “an integer basis of the $\mathbb{Z}$-module $H^*(M)$”?

I thought I understood the concept of a cohomology ring, but am confused by the following statement found in a textbook. Context: M is a symplectic manifold of dimension $2n$. "Let us choose an ...
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34 views

Integrating differential form question

I was given the following question on an exam this morning and was wondering if my solution was correct? The question was "if $\omega = 2xy\,dx+x^2\,dy$ and $C$ an arbitrary curve from $(0,0)$ to ...
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152 views

Principal bundles, connection forms and fundamental vector fields

Suppose $\pi:P\rightarrow M$ is a principal bundle, $\omega\in \Omega^1(P;\mathfrak{g})$ is the connection one form and $\sigma(\cdot)$ is the fundamental vector field associated to some vector field ...
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Petersen Riemannian geometry p86

I'm confused by a computation in Peter Petersen's Riemannian geometry book. We consider $S^{2n+1}$ viewed as embedded in $\mathbb{C}^{n+1}.$ The circle $S^1$ acts naturally on $S^{2n+1}$ by complex ...
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44 views

Question on calculating curvature of a surface given implicitly

I want to find, as an exercise, an expression for the curvature of a surface given by the zero set of a function. I reached a final expression, but when I test it for a sphere I get a non-constant ...
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203 views

Do Carmo :Show a line of curvature C is a plane curve if osculating plane makes a constant angle

Here's the full problem: Assume that the osculating plane of a line of curvature $C \subset S$, which is nowhere tangent to an asymptotic direction, makes a constant angle with the tangent plane of ...
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104 views

Integration on manifolds and improper integration

Consider the usual concept of integral on a smooth manifold (the one built using partitions of unity). When applied to the usual smooth structure of $\mathbb{R}^n$, does it coincide with the concept ...
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117 views

$TS^1$ is Diffeomorphic to $S^1\times \mathbf R$.

I know this is a very basic question. But I am unable to get every detail right. I need to show that $TS^1$ is diffeomorphic to $S^1\times \mathbf R$. (I am using the concept of derivations to ...
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1answer
36 views

What is the tangent space of a two-dimensional domain?

Consider a map $f:M\to N$, and let $p\in M$. We can define the differential of $f$ at point $p$ as a map from $T_pM$ to $T_{f(p)} N$, and this map is linear. And because of that, we can come up with a ...
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Smoothness and algebraicity

My question is the following : given a complex algebraic affine variety $M \subset \mathbb{C}^n$, let us assume that it has moreover a smooth differential structure as a submaifold of ...
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62 views

Curves and tangent vectors in a manifold setting

Consider the following definition: ($M$ denotes a manifold structure, $U$ are subsets of the manifold and $\phi$ the transition functions) Def: A smooth curve in $M$ is a map $\gamma: I \rightarrow ...
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Why don't we write $\nabla_{X}(fY) = f\nabla_{X}Y$ instead of $\nabla_{X}(fY) = f\nabla_{X}Y+ X(f)Y$ for affine connections?

According to do Carmo, in Riemannian Geometry pages 49-50, he says let $\mathcal{X}(M)$ denote the set of all vector fields of class $C^{\infty}$ on $M$. Let $\mathcal{D}(M)$ denote the ring of all ...
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217 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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Complete Riemannian metrics in cylinder $\mathbb{R}\times X$ and cones $\mathbb{R}^{+}\times X$

Consider the cylinder $\mathbb{R}_t\times X$ where $X$ is a compact manifold without boundary. Consider the cylindrical metric $g_{cyl}=g_X+dt^2$. Clearly $(\mathbb{R}_t\times X, g_X+dt^2)$ is a ...
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44 views

Checking alternating tensors

How do I check that $$f(x,y)=x_1y_2-x_2y_1+x_1y_1$$ is an alternating tensor? I did check that f is a tensor, but how do I know if it is alternating by direct computation? Thanks in advance!
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1answer
183 views

Extension Lemma for Smooth maps (Lee vs. Lee)

I've been reading Jeffrey Lee's, Manifolds and Differential Geometry and John Lee's, Introduction to smooth manifolds. In the first book (here, in page 31), after introducing partition of unity, ...
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Parametric equation: tangents

I'm not sure what it's called in English, but a curve can have different types of tangents and if a parametrization of the curve is given by $r(t)$, then if $r'(a) = 0$ and $r''(a) \neq 0$, then at ...
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1answer
48 views

How does a Lie derivative generate a $U(1)$ isometry?

Consider a $2l$-dimensional Riemannian manifold $(M,g)$ without a boundary and let $V=V^{\mu}\frac{\partial}{\partial^{\mu}}$ be a Killing vector field, i.e. $$ \mathcal{L}_Vg_{\mu \nu} = 0 ...
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How do we wedge the complex differentials $\mathrm{d}z^i$ and $\mathrm{d}\bar z^{\bar j}$?

By the standard definition of the wedge product as an alternated tensor product, I would think we have $$\tag{1}\mathrm{d}z^i\wedge\mathrm{d}\bar z^{\bar j}=\mathrm{d}z^i\otimes\mathrm{d}\bar z^{\bar ...
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Complex and Kähler-manifolds

I was woundering if anyone knows any good references about Kähler and complex manifolds? I'm studying supergravity theories and for the simpelest N=1 supergravity we'll get these. Now in the ...
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29 views

Differential Geometry in 3D vs Differential Geometry in n dimensions

I have read (in the introduction to Elementary Differential Geometry by Pressley) that some theorems in 3D differential geometry cannot be extended to differential geometry in n dimensions. Is there a ...
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Curve in a product of tori

Consider the curve $\gamma:\mathbb R\to (\mathbb R/\mathbb Z)^n$ given by $$\gamma(t)=(a_1t,\ldots,a_nt)$$ for generic real numbers $a_1,\ldots,a_n$. Is the image of $\gamma$ dense in $(\mathbb ...
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74 views

A simple question about rational Hodge conjecture

Good evening everyone : In the link here I found the following sentence : The Hodge conjecture predicts that the $\mathbb{Q}$ - linear span of the classes of algebraic subvarities in the cohomology ...
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1answer
44 views

Exterior derivatives involving representations

I have two questions regarding the exterior derivative of vector valued forms when representations are involved: Suppose $V$ is a vector space, $M$ a smooth manifold and $\omega$ is a $V$ valued ...
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46 views

A kind of uniqueness for the double of a manifold

Let $M$ and $N$ be two manifolds with the same boundary. If their doubles $D(M)$ and $D(N)$ are diffeomorphic, are $M$ and $N$ diffeomorphic?
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Tangent space of tangent vector

Let $M$ be a smooth manifold. There's a (split) short exact sequence $$0\to T_aM\to T_v(TM)\stackrel {D_v}{\to} T_aM \to 0,$$ where $v\in TM$ and $a=\pi(v)\in M$. I'm trying to understand what this ...
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4answers
257 views

Examples of familiar, easy-to-visualize manifolds that admit Lie group structures

I have a trouble learning Lie groups --- I have no canonical example to imagine while thinking of a Lie group. When I imagine a manifold it is usually some kind of a $2$D blanket or a circle/curve or ...
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2answers
54 views

Universal covering space of X x classifying space of \pi_1(X)

I am trying to learn about classifying spaces for a Lie group $G$. The question I have is the following: Suppose $X$ is a manifold and $G=\pi_1(X)$ is its fundamental group, is it true that ...
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Quick question about curves and basis

Hello all I have a quick question because I am trying to understand my notes and I am confused. Can anyone here atleast give me a hint or anything! Taking note of the fact that the normal vector, ...
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1answer
42 views

Stereographic projection is conformal in the sense of bilinear forms?

This is a past exam problem from my university. However, the corresponding course sequence does not cover any Riemannian geometry, so I'm not sure how to go about this so much. Let ...
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1answer
53 views

How do geometers define “locally looks like” in differential geometry?

From reading some introductory texts on differential geometry, the author would usually invoke the phrase "locally looks like" when it comes to defining a manifold. For example, the real line is a ...
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65 views

Trivial tangent bundle of manifolds with boundary

In the Lee‘s book there is a proposition stating: If $M$ is a smooth $n$-manifold with or without boundary, and $M$ can be covered by a single smooth chart, then $TM$ is diffeomorphic to $M\times ...
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Concrete example of zero section

I just learnt tangent bundle and I want to get some intuition about zero section (and sections in general). I'm even not clear about what the zero vector is in a tangent space--e.g. just consider ...
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Can someone provide a simple example of the “pre-image theorem” in differential geometry?

I only have a background in engineering calculus. A problem I am currently working on relates to something called a "pre-image theorem" The theorem roughly states: Let $f: N \to R^{m}$ be a ...
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Given a nonzero vector $B$ and a vector-valued function $F$ such that $F(t)\cdot B=t$ for all $t$, show that$F''$ is orthogonal to $F'$

Given a nonzero vector $B$ and a vector-valued function $F$ such that $F(t)\cdot B=t$ for all $t$, and such that the angle between $F'(t)$ and $B$ is constant (independent of $t$). Prove that $F''(t)$ ...
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1answer
92 views

Find the length of a rectifiable curve $a(t)=(t,t^2)$ on $[0,1]$.

Let $a(t)=(t,t^2)$ be defined on $[0,1]$. Put $s_n := \sum_{i=1}^n \frac{1}{n}\sqrt{1+\frac{(2i-1)^2}{n^2}} $. I want to show the exsitence of the limit of $\{s_n\}$ and find its limit without the ...
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92 views

Is there a charaterization of riemannian product manifolds?

Some Riemannian manifolds are expressed as a product manifold. Recently, I have read two articles about space-times. In both articles, the authors prove that a Riemannian manifold $\bar{M}^n$ is ...
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Problem with the concept of connection

I've been told that there is only a canonical way for doing the vertical subspace of the tangent bundle of a manifold and in order to do the horizontal subspace you need a connection. These are very ...
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Isothermal parameterization, Inverse of the Gauss Map

This problem is from Do Carmo's Differential Geometry of Curves and Surfaces. It is question 13 from chapter 3.5, to be specific. Suppose that S is a minimal surface without any umbilical points ...