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)

1
vote
0answers
20 views

Map of smooth manifolds

Let $M$ and $N$ be smooth, connected $n$-dimensional manifolds. Let $M$ be compact and non-empty. Show that every embedding $f: M \to N$ is a diffeomorphism. So because $f$ is a embedding we have ...
0
votes
0answers
13 views

Derivative group action

Let $\phi : G \times M \rightarrow M$ be a group action on a smooth manifold $M$ and Lie group $G$. Then we define $$f(t):=\phi(g(t),d(t)).$$ Now I'd say: $$f'(t) = D\phi(g(t),d(t))(g'(t),d'(t)).$$ ...
1
vote
2answers
67 views

On a scale of 1 to 10 how far is this manifold from being a normal bundle?

(DIS)-CLAIMER: All the manifolds considered in this post are completely humble and have no additional structure beyond the smooth structure. Let $Y$ be a submanifold of $M$ and let $(-)^0$ denote the ...
2
votes
0answers
23 views

Advanced calculus: Solving quaternion differential equations

I have a system of two differential equations $$\frac{\partial X(t)}{\partial t}=a_1 A X(t)+a_2X(t) B+a_3 C Y(t)+a_4Y(t) D+a_5$$ $$\frac{\partial Y(t)}{\partial t}=b_1 E X(t)+b_2X(t) F+b_3 G ...
-2
votes
0answers
19 views

What is the derivative of operator ? and what is degree of operator?

What is the derivative of operator ? and what is degree of operator?and what is symbol of operator? I encounter these ideas in chapter 4 of Hamilton's THREE_MANIFOLDS WITH POSITIVE RICCI CURVATURES. ...
1
vote
0answers
24 views

Integral curves on immersed submanifold

An exercise of the book "Introduction to smooth manifolds - John M. Lee" asks to prove that if $S$ is a closed embedded submanifold of a manifold $M$, and $X$ is a vector field on $M$ tangent to $S$, ...
2
votes
0answers
12 views

The lie bracket as a limit

Let $M$ be a manifold and $f_t$ be the monoparametric group of local transformations generated by a vector field $X$. Suppose $w$ is a $k$-form and $x_1,\dots,x_k$ are vector fields. How can we see ...
1
vote
0answers
27 views

a question about Fenchel's theorem(differential geometry)

I am an undergraduate student studying differential geometry right now. I am just finishing reading how to prove Fenchel's theorem:The total curvature of a smooth closed curve in 3-dimensional space ...
1
vote
1answer
32 views

second fundamental form and connection forms

I am reading this paper that has the following: Suppose $M$ is an (n-1) dimensional closed hyper surface immersed in $\mathbb{R}^{n}$. Let $e_1, \cdots, e_n$ be orthonormal frame in $\mathbb{R}^n$ ...
5
votes
0answers
45 views

how much differential structure can we put on countable manifolds?

The motivation for this question is that I would like to formulate Lagrangian mechanics in a purely discrete setting (see also my older question at physics.se). Unfortunately several key pieces of ...
2
votes
0answers
25 views

Let $p=(5,0,-4)$ and $v \in T_{(5,0,-4)}M$. Compute $(F^{*}\omega)_p(v)$.

Let me show my work before presenting the problem itself. Let $M=\{(x,y,z) \in \mathbb{R}^3 : x+y=5, x+z=cos^2y\}$. We can easily see that $M$ is a submanifold of $\mathbb{R}^3$ of dimension $1$. ...
1
vote
2answers
36 views

What am I doing wrong? Derivative of pedal

Let $\gamma$ be a unit speed curve $\gamma : I \to \mathbb R^2$. The pedal is given by $P (s) = (\gamma (s) \cdot N(s)) N(s)$. I tried to calculate the derivative as follows: $$ P' = (\gamma N)' N ...
4
votes
1answer
38 views

Parametrizations and coordinates in differential geometry - what's the difference?

From what I've read one can introduce the notion of a tangent vector to a point on a manifold in terms of an equivalence class of curves passing through that point (the equivalence relation being that ...
0
votes
0answers
34 views

Natural derivative of Vector Fields on manifolds

I'm learning about connections and my book says that there is no natural derivative for a vector field on a manifold. Wouldn't it be possible to cook up a connection by just letting $\nabla_{v_p}X = ...
1
vote
1answer
54 views

Connection between harmonic functions, Bochner Laplacian and Ricci curvature

I stumbled upon the following claim in a paper: "We write the (Bochner) Laplacian in suffix notation: $\Delta_B = \nabla ^k \nabla_k$". after this statement, the following is written: ($M$ is a ...
0
votes
0answers
17 views

Normal Vectors to Action of Orthogonal Group

Let $X\in\mathbb{R}^{n\times r}$ be a fixed matrix with orthogonal columns, and let $U\in\mathbb{R}^{n\times r}$ be given. Because the group of orthogonal $r\times r$ matrices, $O(r)$, is a compact ...
4
votes
1answer
28 views

Making a bijection into a diffeomorphism

Given a set $M$, one that can be made into a smooth manifold, and a bijection $f:M\to M$, does there exist a differentiable structure on $M$ such that $f$ is a diffeomorphism? In case it's not always ...
1
vote
3answers
188 views

Is a ball noncompact?

A compact manifold usually refers to "a manifold without a boundary", for example the usual 2-sphere $S^2$. What about a manifold with a boundary? Intuitively, I think such an example, e.g. a ball ...
2
votes
1answer
38 views

Unit sphere and Ricci curvature

Why is it that on the unit sphere the Ricci curvature Ric = g (where g is the metric defined on the unit sphere) ?
0
votes
1answer
29 views

Determine the lines of curvature of $z=xy$

I have to find the lines of curvature of $z=xy$ I calculate Weingarten Matrix as described below $p_u = (1, 0, v),p_v=(0,1,u),\nu =\frac{1}{\sqrt{1+u^2+v^2}}(-v,-u,1)$ so, $E=1+v^2,F=uv,G=1+u^2$ ...
5
votes
2answers
176 views

Nonexistence of local isometry between equidimensional Riemannian manifolds

Recall that all inner product spaces of the same dimension are isometric. For example, if $(M,\mathrm{g})$ and $(N,\mathrm{h})$ are Riemannian manifolds of the same dimension, then ...
0
votes
1answer
40 views

Extension of vector to vector field and curvature two-form

Let $(P,\pi, M)$ be a principal bundle with structure group $G$ and let $\omega$ be a connection on this bundle. The curvature two-form is $\Omega = D\omega$ and it's quite easy to show that ...
2
votes
1answer
33 views

Compute in the chosen charts of $M$ and $S^1$ the expression of $DF_{(5,0,-4)}$

Let me show my work before presenting the problem itself. Let $M=\{(x,y,z) \in \mathbb{R}^3 : x+y=5 x+z=cos^2y\}$. We can easily see that $M$ is a submanifold of $\mathbb{R}^3$ of dimension $1$. We ...
2
votes
0answers
27 views

Action of $H^1$ on spin structures

Since the set of spin structures on a principal $SO(n)$-bundle on a manifold $X$ is in one to one correspondence with $H^1(X,Z_2)$, this group admits an action on spin structures. I wanted to know if ...
0
votes
1answer
28 views

Lagrange multipliers and critical points (differential form description).

On $M \times V^*$, where $M$ is a differentiable manifold (not necessarily equipped with a metric) and $V^*$ is dual to a vector space $V$, one can define a Lagrange function $F = f +v^*h(x)$ using ...
4
votes
1answer
30 views

What is the difference between Nakano Postivity and Griffiths Positivity of Hermitian vector bundles?

I am currently reading "Complex Differential Geometry" by FY Zheng on the curvature of Hermitian vector bundles. In section 7.5, he described a Hermitian vector bundle $(E,h)$ over a complex manifold ...
0
votes
1answer
24 views

When do we call the boundary $\partial\Omega$ of a bounded domain $\Omega\subseteq\mathbb{R}^n$ smooth?

When do we call the boundary $\partial\Omega$ of a bounded domain $\Omega\subseteq\mathbb{R}^n$ smooth? I can't find a formal definition. I know, that we say, that $\partial\Omega$ has a ...
4
votes
1answer
25 views

principal curvature of the flat torus

I am looking at the Hopf-fibration and I am looking at the preimage of the equator in $\mathbb{S}^2$. I think that I have proved that this is just the flat torus and now I want to calculate the ...
1
vote
0answers
34 views

Injectivity of the Differential of Smooth Map

I am trying to answer the following question: Let $M = \{(x,y)\in \mathbf{R}^2 : x^2 + y^2 < 1\}$. Define a smooth or $C^\infty$ function by $f\colon M \rightarrow \mathbf{R}^2$ as ...
1
vote
0answers
32 views

Kinds of isometries preserving the curvature tensor

We talked about isometries /local isometries and linear isometries in the lecture, but unfortunately we did not say when the isometry preserves the curvature tensor or sectional curvature. First I am ...
0
votes
1answer
28 views

Tensor Product of Vectors

let $S,T$ be respectively $k-, n- $ tensors; $k,n>0$. Then we define the tensor product $$ T \otimes S(x_1,x_2,....,x_{k+n}):=T(x_1,...,x_k)S(x_{k+1},...,x_{k+n}) $$ (their product as Real numbers, ...
2
votes
1answer
40 views

Tangent space as derivations exercise

Thinking of the tangent space to a manifold as derivations is a concept which just kind of eludes me. I am comfortable thinking about tangent vectors as equivalence classes of curves and with the ...
0
votes
1answer
24 views

Manifold with boundary given as the pre-image of a subset of $\mathbb{H}^n$

Let $f \colon \mathbb{R}^{n+k} \to \mathbb{R}^n$ be a function of class $C^r$ for $r>1$. If $M = f^{-1}(0)$ and $0$ is a regular value of $f$, then we know (using implicity functions theorem) that ...
3
votes
2answers
115 views

What's my mistake in the calculation?

Summation convention holds. If $\frac{\partial}{\partial t}g_{ij}=\frac{2}{n}rg_{ij}-2R_{ij}$, then ,I compute: $$ \frac{1}{2}g^{ij}\frac{\partial}{\partial ...
3
votes
1answer
26 views

Connection form on a frame bundle

Let $(E(M),\pi,M)$ be the frame bundle over a manifold $M$ of rank $n$. Consider a covering of $M$ by open neighborhoods $U_{\alpha}$. Let $s_{i}$ and $t_{j}$ where $i,j\in {1,...,n}$ be frames of $M$ ...
3
votes
2answers
76 views

Gaussian Curvature of $x^4+y^4+z^4=1$

Let $S=\{(x,y,z)\in \mathbf R^3 | x^4+y^4+z^4=1 \}$ . To compute the Gaussian curvature $k$ of $S$, I tried an elementary method to find $dN_p$. Let $\alpha (t) = (x(t),y(t),z(t))$ be an parametried ...
0
votes
1answer
21 views

$M\subset\mathbb{R}^n$ is a open subset, $p\in M$ is arbitrary. Find $T_pM$ and $N_pM$

PROBLEM: $M\subset\mathbb{R}^n$ is a open subset, $p\in M$ is arbitrary. Find $T_pM$ and $N_pM$. I know how to determine $T_pM$ and $N_pM$ for explicit examples, but I dont know how to handle this ...
1
vote
0answers
21 views

Vector field from group action

Let $\Phi: G \times \mathbb{R}^4 \rightarrow \mathbb{R}^4$ be a group action where $G = \mathbb{R}/(2 \pi \mathbb{Z}).$ Then $$\Phi(\theta,(x_1,x_2,p_1,p_2)) = ( R(\theta) (x_1,x_2)^T, R(\theta) ...
3
votes
2answers
76 views

Boundedness of the norm of the Riemann curvature tensor

Let $(M,g)$ be a Riemannian manifold and let $R(X,Y)Z$ be its $(3,1)$ Riemann curvature tensor given by $$R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z$$ Let the input vectors $X,Y,Z$ ...
2
votes
1answer
55 views

Can't we really add together two points on a manifold?

Let us consider a classical mechanical system with observables being smooth functions $C^\infty(X)$ on a Poisson manifold $X$. The algebra of observables will be denoted as $A$ Next we can define ...
8
votes
4answers
605 views

Mathematics is not a spectator's sport? [on hold]

The title is a sentence by John M. Lee, from his book "Introduction to Topological Manifolds". Indeed, I was wondering if one can learn mathematics in a passive way, just reading the books and ...
3
votes
1answer
25 views

Existence of a vector field which dominates the first local vector fields given by the charts of a locally finite covering

Let $M$ be a smooth manifold, let $\{U_i,\psi_i\}_{i\in I}$ be locally finite family of charts and let $K_i\subseteq U_i$ be compact subsets. Does there exist a vector field $X$ on $M$, such that ...
1
vote
1answer
36 views

Show that $M$ is a differentiable submanifold

Problem. Let $f_i:\Bbb{R}^4\to \Bbb{R}, \,\, i=1,2,3,$ be defined by $$f_1(x_1,x_2,x_3,x_4) = x_1x_3-x_2^2\\f_2(x_1,x_2,x_3,x_4)=x_2x_4-x_3^2\\f_3(x_1,x_2,x_3,x_4)=x_1x_4-x_2x_3.$$ Then $M=\{x\in ...
2
votes
0answers
21 views

Local parametrizations and coordinate charts on manifolds

I have recently had discussions on related questions about coordinate charts on here which has started to clear up some issues in my understanding of manifolds. Apologies in advance for the ...
4
votes
1answer
53 views

Topological information from metric tensor

Suppose I am working with a Riemannian manifold $(M,g)$, and I have a particular coordinate expression for the metric $g$. What topological information can I infer about the manifold $M$? For ...
1
vote
0answers
65 views

A variational approach to symplectomorphisms

Let $(\Sigma,\omega)$ be a compact symplectic surface, and let $\mathcal{D}$ be the group of diffeomorphisms of $\Sigma$. Is there some known variational approach to determine if an element ...
3
votes
2answers
26 views

Stereographic projection and lengths

We know the Stereographic Projection doesn't preserve areas except for the points on the plane such that $x^2+y^2=1$, because that's where $dA=dxdy$. I was wondering what would happen with lengths: ...
2
votes
1answer
23 views

Integral of Laplace-Beltrami operator over a manifold

Consider an equation $$\Delta u=-he^{u}$$ over a compact 2-manifold $M$, where $u\in C^{\infty}(M)$. In paper "Curvature functions for Compact 2-Manifolds" by Kazdan&Warner it is said that ...
-1
votes
1answer
44 views

Invariance of volume along Ricci flow

The Riemannian metric is $g_{ij}$,its inverse is $g^{ij}$,and the induced measure is $du=u(x)du$ where $u(x)=\sqrt{det(g_{ij})}$.The scalar curvature is $R=g^{ij}R_{ij}$ . $r=\frac{\int R du}{\int ...
1
vote
1answer
92 views

Some possible mistakes in Bott and Tu

I've been reading Bott and Tu's book "Differential Forms in Algebraic Topology" and it seems that this book has an enormous list of errors. Therefore I would like to confirm if the parts that I will ...