# Tagged Questions

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 ...

20 views

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 ...
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 ...
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 $i:M \rightarrow N$ ...
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 ...
77 views

### 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 ...
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 ...
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 ...
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 ...
115 views

### 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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
87 views

### 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 ...
62 views

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 ...
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 ...
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?
91 views

### 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 ...
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 ...
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 ...
33 views

### 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, ...
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 ...
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 ...
65 views