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 ...

learn more… | top users | synonyms (1)

3
votes
1answer
104 views

Global sections of holomorphic vector bundles

Let $X$ be a complex manifold, and $\mathbb{L}\rightarrow X$ a holomorphic line bundle over $X.$ Can we always find global sections of $\mathbb{L}$? (other from the one that's identically zero) On a ...
1
vote
0answers
45 views

Why are free objects related to tensors important?

Let $R$ be a commutative ring and $M$ be an $R$-module. Then, we can construct the tensor algebra $T(M):=\oplus_{n\in \mathbb{N}} M^{\otimes n}$ which is likely a free object. That is, for any ...
5
votes
1answer
59 views

Intuitive explanation of the term manifold

I am reading Christopher Bishop's "Pattern Recognition and Machine Learning" and in the first chapter, where he talks about the curse of dimensionality, he gives the following example: Consider, ...
3
votes
1answer
138 views

Grassmannian a Fano manifold?

I am interested in answering the following: Is it true that the Grassmannian $G(k,\mathbb{C}^n)$ is a Fano manifold? How can I see if its anticanonical bundle is ample?
1
vote
2answers
67 views

Analytic hypersurface as union of irreducibles

Let $X$ be a complex manifold. Then any analytic subvariety $V$ of codimension 1 (that is, any analytic hypersurface) can be expressed uniquely as the union of irreducible analytic hypersurfaces ...
0
votes
1answer
70 views

What is the linear series $|mL|$?

I am studying complex geometry and I am trying to find out what is the definition of the linear series $|mL|,$ where $L$ be a line bundle over a compact Kahler manifold $X^n.$ In particular, I know ...
2
votes
1answer
580 views

Metric on n-sphere in terms of stereographic projection coordinates

The metric on the $n$-sphere is the metric induced from the ambient Euclidean metric. Find the metric, $d\Omega^2_n$, on the $n$-sphere and the volume form, $\Omega_{S_n}$ , of $S^n$ in terms of the ...
2
votes
1answer
145 views

Normal of a coons patch at a given point

Disclamer: Rendering the Coons patch is part of 3D Graphics homework, but finding the normals at a given point isn't. Just curious. Here's what I got so far: It's a Coons patch defined by four ...
0
votes
0answers
79 views

show that if $M \times \mathbb{R}^{n}$ is orientable than so is $M$

I need to show if $M \times \mathbb{R}^{n}$ is orientable than so is $M$, where $M$ is connected manifold. $R^{n}$ has standard orientation (determined by standard basis ) and by the assumption $M ...
0
votes
1answer
60 views

Gradient of a function defined on a surface

Let $V:R^{3}\rightarrow R$ be a differential function. Let $$A= \begin{pmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \\ \end{pmatrix}. ...
0
votes
1answer
90 views

Are all 2D tensors in a specified flat metric equal to that same metric conformally scaled?

I have a tensor $T_{mn}$ where its indices coorespond to a flat metric $g_{mn}$. I want $T_{mn}$ to be a new metric $\tilde{g}_{mn}$, such that $T_{mn}(g_{rs}) = \tilde{g}_{mn}$. A theorem says that ...
5
votes
0answers
111 views

What's wrong with my osculating paraboloid?

I'm trying to learn about the Second Fundamental Form and I thought it would be fun to set up a surface in Geogebra and try to calculate the osculating paraboloid as I moved a point around on it, for ...
3
votes
1answer
87 views

Two definitions of curvature

my question is about the compatibility of two definitions of curvature of a Riemannian manifold. In particular I refer to the one from algebraic geometry and the one from differential geometry. ...
4
votes
1answer
78 views

Fibered product of Hopf Fibrations

I am wondering: what is the vector bundle associated to the Hopf-fibration $S^{3}\to S^{2}$? What is the fibered product of two Hopf-fibrations? Thanks.
1
vote
1answer
85 views

Proof that the Gaussian Curvature is a Ratio of Areas

Let $S \subset \mathbb{R}^3 $ be a smooth surface, and let $S^"$ be the unit sphere, and let $n: S \to S^2$ be a given Gauss map. I want to prove that the Gaussian curvature $K(p)$ at a point $p \in ...
3
votes
1answer
53 views

Why is $\ker\omega$ integrable iff $\omega\wedge d\omega=0$?

Suppose $\omega$ is a nonvanishing $1$-form on a $3$-manifold $M$. It's known that $\ker\omega$ is an integral distribution iff $\omega\wedge d\omega=0$. I'm trying to understand this, but I don't ...
3
votes
2answers
62 views

What does $[L]=[I]^{-1}[II]$ mean?

I have a question about one of the equations in my notes. Matrix representations of Weingarton map, first fundamental form and second fundamental form satisfies $[L]=[I]^{-1}[II]$ According to ...
1
vote
0answers
36 views

Fibered product of symplectic fiber bundles

Let $P_{1}\to B$ and $P_{2}\to B$ be smooth fiber bundles over the same base manifold $B$, and let us assume that the total spaces $(P_{1},\omega_{1})$ and $(P_{2},\omega_{2})$ are symplectic ...
2
votes
0answers
111 views

Analogue of the Euler Class of a Circle Bundle and the Global Angular Form

This is a general question that asks whether there is geometric significant to differential forms that restrict to G-invariant volume elements on the fibers of a principal G-bundle. For an SO(2) ...
2
votes
0answers
55 views

Help! How to derive the result related to Darboux derivative?

First, define Darboux derivative. There is one Lie group $G$ and one manifold $M$. Let $\phi:M\rightarrow G$ be a smooth map. The Darboux derivative $\Delta(\phi):TM \rightarrow M\times \mathfrak g$ ...
7
votes
3answers
302 views

What is the intuition behind the definition of the differential of a function?

What is the intuition behind the definition of a differential of a function in differential geometry? i.e. $$df(p)(v_{p}) =v_{p} (f)(p) $$ where $v_{p} \in T_{p} M$ is a vector in the tangent space to ...
2
votes
1answer
202 views

Integration on manifolds with singular points, corners

I'm looking for interesting examples of application of Stokes theorem for manifolds with singularities/corners. The theorem was mentioned here: ...
1
vote
1answer
40 views

requirement of openess of the subset of a manifold for mayer-vietoris theorem

So for a given manifold $M$ and two of $U,V$ open sets that can be used to cover $M$, one can use to mayer-vietoris theorem to relate the decomposed de rham cohomology of $M$ with that of $U$ and $V$. ...
19
votes
4answers
580 views

What is a Manifold?

Now we always encounter definition of a manifold from a mathematical point of view where it is a topological space along with a family of open sets that covers it and the same old symphony. My ...
7
votes
3answers
308 views

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 ...
1
vote
1answer
20 views

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 ...
1
vote
1answer
63 views

Prove that an isolated point of $C: (f=0)\subset \mathbb{R}^2$ must be a max or min of $f: \mathbb{R}^2 \rightarrow \mathbb{R}$.

Let $f\in \mathbb{R}[x,y]$ and let $C: (f=0)\subset \mathbb{R}^2$; we say that $P\in C$ is isolated if there is an $\epsilon >0$ such that $C\cap B(P,\epsilon)=P$. Prove that if $P\in C$ is an ...
0
votes
0answers
28 views

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.
6
votes
1answer
151 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 ...
0
votes
1answer
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 ...
4
votes
1answer
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 ...
1
vote
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 $i:M \rightarrow N$ ...
6
votes
1answer
153 views

Proving that given any two points in a connected manifold, there exists a diffeomorphism taking one to the other

Suppose $M$ be a connected manifold and $x, y \in M$ are two points. Then I'm trying to show that there is a diffeomeorphism $f$ of $M$ that takes $x$ to $y$. Since the set of points for which there ...
3
votes
1answer
222 views

Proving that the quotient manifold is orientable if and only if the group action is orientation-preserving

I'm trying to solve the following exercise in Lee's book. Suppose M is a connected, oriented smooth manifold and Γ is a discrete group acting freely and properly on M. We say the action is ...
0
votes
2answers
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 ...
2
votes
1answer
49 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 ...
2
votes
0answers
116 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 ...
1
vote
1answer
61 views

Fundamental group of a compact hyperbolic manifold

Let $M$ be a compact hyperbolic manifold, and $\tilde M = H^n$ the universal covering. Now let $\Gamma$ be the group of Decktransformations. So we have $\tilde M / \Gamma = M$. My question: Is it ...
1
vote
2answers
80 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 ...
1
vote
2answers
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 ...
1
vote
2answers
62 views

The bundle vector $f^\ast(\xi)$ for Moebius over $S^1$

Take the Moebius band like a vector bundle $\xi$ over the circle $S^1$ and the functions $f_n(z)=z^n$ then my question is: how describe the vector bundle define for the pullback $f^\ast(\xi)$ for ...
0
votes
1answer
211 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 ...
1
vote
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 ...
2
votes
1answer
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 ...
2
votes
0answers
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 ...
4
votes
1answer
107 views

Is there an intuitve motivation for the wedge product in differential geometry?

I've recently started studying differential forms and have been looking at differential forms. I'm struggling to understand the motivation for introducing the notion of the wedge product. Does it ...
3
votes
2answers
118 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 ...
9
votes
3answers
200 views

What is the relation between dx in elementary calculus and dx in differential geometry?

I've recently started studying differential geometry and was really hoping that in doing so I'd finally have an answer to something that's been bugging me since I first learnt calculus - what is ...
1
vote
1answer
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 ...
2
votes
0answers
99 views

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 ...