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)

13
votes
1answer
258 views

Can we generalize the regular value theorem even beyond the Ehresmann's theorem?

The formulation is complicated, but the answer may be some clever usage of the partition of unity, because locally the answer is given by the regular value theorem and the whole problem is to glue it ...
10
votes
1answer
188 views

Invariant submanifolds

Let $M$ be a smooth manifold, and let $N$ be a submanifold. Let $V$ be a smooth vector field on $M$ which generates a flow $\Phi_t$ on $M$. My intuition tells me (perhaps modulo some technical ...
9
votes
1answer
548 views

Geometric intuition for the Weingarten map

Parameterize a hypersurface $M$ by $r: \Omega \rightarrow \mathbb{R}^n$, and let $T_p M$ denote the tangent space at $p = r(u)$. We define the Weingarten map to be the linear map $L_p : T_p M ...
8
votes
1answer
120 views

What is wrong with this exercise in do Carmo's Differential Geometry?

This is an exercise in do Carmo's Differential Geometry: Let $\alpha : I \longrightarrow S$ be a curve parametrized by arc length $s$, with nonzero curvature. Consider the parametrized surface ...
6
votes
1answer
71 views

Why the connected sum is a differentiable manifold

maybe this is a stupid question. I know how to prove that the connected sum $M\#N$ of two topological manifolds is a topological manifold, however I don´t know how to prove that the connected sum of ...
6
votes
1answer
147 views

Lie derivative of curvature

Let $M$ be a Kähler manifold, with Kähler metric $g$. Let $X$ be a holomorphic Killing vector field of $g$, i.e. $\mathcal{L}_{X} g = 0$, where $\mathcal{L}_{X}$ is the Lie derivative along $X$. Let ...
6
votes
1answer
638 views

Want to learn differential geometry and want the sheaf perspective

I would like to learn some differential geometry: basically manifolds, differentiable manifolds, smooth manifolds, De Rham cohomology and everything else that is pretty much part of a course in ...
5
votes
1answer
142 views

Complex differential geometric form of the Grothendieck–Hirzebruch–Riemann–Roch theorem

From the wikipedia article, it seems that there should be a differential geometric form of the Grothendieck-Riemann-Roch theorem with schemes replaced by complex manifolds and quasi-coherent sheaves ...
5
votes
1answer
86 views

Detecting the genus of a surface by walking straight ahead

I have a - maybe misled - intuition that one could "somehow" detect that one lives on a torus (of genus 1) just by walking around straight ahead and making appropriate observations. Living on a ...
5
votes
1answer
154 views

special covering of a non-compact manifold

I'm very stuck on the following exercise in the book "A Comprehensive Introduction to Differential Geometry V.1" by Michael Spivak: Let $M^m$ be a smooth connected non-compact manifold. Show that $M$ ...
4
votes
1answer
246 views

Tensors as mutlilinear maps

I am aware that many books on differential geometry define tensors as multilinear maps. Namely $$ V\otimes W := L_2(V^*\times W^*,\Bbb F) $$ I am also aware that this space is isomorphic to the tensor ...
4
votes
1answer
494 views

approximating geodesic distances on the sphere by euclidean distances of a transformed sphere

Is there a way to find a function $F:\mathbb S^2 \rightarrow \mathbb R^3$ of class $C^1$, minimizing $$\int_{\mathbb S^2\times\mathbb S^2}(d(F(x),F(y))−\delta(x,y))^2 dx dy$$ , where $d$ stands for ...
4
votes
1answer
397 views

local diffeomorphism

I hope this finds you all well. Could someone give me an example of a local diffeomorphism from $\mathbb{R}^p$ to $\mathbb{R}^p$ (function of class say $C^k$ with an invertible differential map in ...
4
votes
1answer
177 views

How do you show that the Laplacian is the square of the (Euclidean) Dirac operator?

If I understand correctly, the Euclidean Dirac operator is given by $$D=\sum_{i=1}^n e_i \frac{\partial}{\partial x_i},$$ where $e_i$ are bases for $Cl_{0,n}(\mathbb{R})$, i.e., the $n$-dimensional ...
3
votes
1answer
50 views

Normal bundle of the two-dimensional sphere manifold embedded in $\mathbb R^4$

Let $M \subset \mathbb R^4$ be a smooth manifold diffeomorphic to $S^2$. How can one prove that normal bundle of $M$ has at least one non-vanishing global section. I think that $M$ should be ...
3
votes
1answer
43 views

Diffeomorphic connected hypersurfaces

Given a four dimensional Lorentzian manifold $\mathcal{M}$ (a manifold with a metric $g_{\mu\nu}$ in the tangent bundle with signature (-1, 1, 1, 1)), we define a global spatial foliation by a ...
3
votes
1answer
55 views

tangent/normal space to set of symmetric isospectral matrices

Let $\Lambda = \{\lambda_1, \ldots, \lambda_n\}$ be a set of $n$ distinct real numbers. $M_n(\mathbb{R})$ denotes the set of all $n \times n$ real matrices, and for $B\in M_n(\mathbb{R})$, $B^T$ ...
3
votes
1answer
92 views

Who invented the Riemann Sphere?

I have seen suggested that someone other than Riemann first came up with the Riemann Sphere. Is this correct? And if so, who did invent it?
3
votes
1answer
38 views

Boundedness of the operator $[\Lambda, \Theta]$.

I am reading Griffiths-Harris book on algebraic geometry and in "Theorem B", where he proves (the analogue of Serre's vanishing theorem) that Let $M$ be a compact, complex manifold, $L\rightarrow ...
3
votes
1answer
40 views

Why is the restricted holonomy the identity component of the holonomy group?

Let $M$ be a connected smooth paracompact manifold, $E$ a vector bundle over $M$ with fibre $\mathbb R^k$, and $\nabla$ a connection on $E$. It is known that Hol$^0(\nabla)$ is a connected Lie ...
3
votes
1answer
74 views

de Rham cohomology of $\mathbb{R}P^n$ via action by $SO(m+1)$

In lecture, my teacher proved the theorem that given a smooth $G$-action by a compact, connected Lie group on a manifold $M$, the de Rham cohomology of the $G$-invariant differential forms $H^p_G(M)$ ...
3
votes
1answer
68 views

Differential Geometry review questions. Need help

I have a final coming up in Differential Geometry and we got a review worksheet and I am having serious trouble with two problems. I'm still chugging along at them but I need help understanding. I ...
3
votes
1answer
76 views

motivating the conservation of symplectic area by way of general (coordinate) covariance

I'm trying to motivate why a symplectic structure captures exactly the right structure one needs to do classical mechanics. The easiest part of this story goes like this: we need a procedure for ...
3
votes
1answer
127 views

Basic question about definition of Chern classes

Apologies if there is something I missed with a quick internet search, but why do we define Chern classes for complex vector bundles (instead of real vector bundles for example)? If we define chern ...
3
votes
1answer
65 views

Manifold of fixed points

Let $M$ be a smooth manifold and let $G$ be a Lie group smoothly acting on $M$. Then, under suitable assumptions (if $G$ acts freely and properly on $M$) we have a new smooth manifold $M/G$ ...
2
votes
1answer
51 views

curve integral - intersection between plane and sphere

I am going to calculate the line integral $$ \int_\gamma z^4dx+x^2dy+y^8dz,$$ where $\gamma$ is the intersection of the plane $y+z=1$ with the sphere $x^2+y^2+z^2=1$, $x \geq 0$, with the orientation ...
2
votes
1answer
90 views

Moment map of the action of $SU(2)$ on $\mathbb C^{2n}$

Let $SU(2)$ acts on symplectic space $((\mathbb C^2 -\ (0,0))^{n},\omega)$, where $$\omega=dx_1\wedge dx_2+dx_3\wedge dx_4+\cdots+dx_{4n-3}\wedge dx_{4n-2}+dx_{4n-1}\wedge dx_{4n}$$ as ...
2
votes
1answer
30 views

Gauss curvature using metric and Riemannian curvature

I learnt that the Gauss curvature is given by: $$K = \frac {eg - f^2}{EG - F^2}$$ where $E, F, G$ are coefficients of the first fundamental form and $e,f, g$ are coefficients of the second ...
2
votes
1answer
33 views

Tangent line to a differentiable curve

Let $\alpha: I \to \mathbb{R}^{n}$ be a differentiable curve such that $\alpha'(a) \neq 0$ for some $a \in I$. The line $L \subset \mathbb{R}^{n}$ through $\alpha(a)$ is the tangent line to the curve ...
2
votes
1answer
116 views

Show isometry of flow on a compact Riemannian manifold where the vector field is Killing

Let $(M,g)$ be a Riemannian manifold, $\nabla$ the Levi-Civita connection of $g$. A vector filed $V$ on $M$ is called a Killing field if for every $p\in M$ and every $X,Y\in T_p M$, $$ g(\nabla_X V, ...
2
votes
1answer
37 views

Embedded and Non-Parametric Surface definition

What does it mean for a minimal surface to be embedded? For example the Scherk surfaces? How would I define what 'an embedded surface' is? And also what does it mean for a surface to be ...
2
votes
1answer
30 views

surface curvature

I would like to proof the existence or the non-existence of a finite surface which has 2 different radius of curvature $R_1$ and $R_2$ that are: constant on the whole surface finite different each ...
2
votes
1answer
78 views

Tangent space to a surface at boundary points

Let $M$ be a $2$-dimensional compact oriented surface in $\mathbb R^3$ with boundary $\partial M$. For any $p\in M \setminus \partial M$ tangent vectors are defined as speed vectors of smooth curves ...
2
votes
1answer
121 views

Geodesic of Elliptic Hyperboloidv

I have the set $$M=\{(x,y,z)\in\mathbb{R}^3:z^2-(a^2x^2+b^2y^2)=R^2,\ z>0\}$$ and I have to write the differential equation that describe the geodesic curve and draw it. I used the ...
2
votes
1answer
52 views

vector field on $\mathbb{R} P^2$

Actually this is a quesion in Lee's book, Manifolds and differential geometry. I have problems working with projective spaces as manifolds.(e.g. what are curves in projective spaces ? I need to know ...
2
votes
1answer
55 views

When does the difference between a vector bundle and the associated frame bundle matter?

In the comments to this question How a principal bundle and the associated vector bundle determine each other, it was remarked that while there is a bijective correspondence between rank $n$ vector ...
2
votes
1answer
54 views

Signed curvature and turning angle

I am trying to find the signed curvature of a function, I have so far that $$g'(t)=(\cos(\cosh(t)), \sin(\cosh(t)))$$ I know that $g'$ is unit speed so i don't have to parametrize by arc length, and ...
2
votes
1answer
97 views

Showing that a subset of the real projective plane is a smooth manifold under given condition

I'm trying to solve exercise 9.7 in Tu's introduction to manifolds: Let $F(x_{0},x_{1},x_{2})$ be a homogeneous polynomial of degree $k$. Consider the homogeneous coordinates ...
2
votes
1answer
138 views

Question about Lie derivative

$(M,w)$ is symplectic manifold. $f_t : M\to M$ is a symplectic isotopy between $f_0=id$ and $f_1$. Let X_t be the vector field on M satisfying $d(f_t)/dt=X_t(f_t)$ Now I differentiate $(f_t)^*w$. Here ...
2
votes
1answer
64 views

Relation between $L^1(\partial\Omega)$ and the surface integral on $C^1$ domains

Let $\Omega \subset \mathbb{R}^n$ be a $C^{1}$ bounded domain. It is possible to define the space $L^1(\partial\Omega)$ as the set of functions $u\colon \partial\Omega \to \mathbb{R}$ with the finite ...
2
votes
1answer
33 views

isometric imbedding of the projective plane

How to isometrically imbed the projective plane (identifying antipodal points of the unit sphere) in $\mathbb{R^5}$? My textbook indicates that there is an isometric imbedding of the projective ...
2
votes
1answer
124 views

Pullback distributes over wedge product

I'm looking to prove that the pullback of a smooth function distributes over wedge product, i.e. $$\varphi^*(\omega \wedge \eta) = \varphi^* \omega \wedge \varphi^* \eta. $$ Here the question is ...
2
votes
1answer
119 views

Riemann tensor symmetries

The Riemann tensor has its component expression: ...
2
votes
1answer
91 views

[ANSWERED]Lie brackets on vector fields

We consider $v = \frac{\partial}{\partial x}$ and $w = x * \frac{\partial}{\partial z} + \frac{\partial}{\partial y}$. I need to first find the Lie bracket between them which i get to be: ...
2
votes
1answer
109 views

Why is this a differentiable structure on the product manifold?

Suppose $M$ en $N$ are differentiable manifolds with differentiable structures $\{(U_a,x_a)\}$ and $\{(V_b,x_b)\}$ resp. Consider $M\times N$ and the mappings $z_{ab}(p,q):=(x_a(p),y_b(q))$ with $p\in ...
2
votes
1answer
77 views

Tangent map of the inclusion map of a submanifold

I'm wondering if anyone could help me with the following question let $M$ be the Minkowski spacetime, let $f\in C^{\infty}(M) ; f(m)=x^{0}(m)$, with $\{x^{\mu}\}$ being a global Cartesian coordinates ...
2
votes
1answer
129 views

Torsion and Non-metricity Tensor on a Surface

In differential geometry of surfaces, how can one define a non-zero Torsion tensor? It seems that the connection you provide has always to be symmetric since, by definition, ...
2
votes
1answer
89 views

Confusion over notation in a book on the mathematics of QFT by Faria-Melo

While formulating this question, I arrived at a likely interpretation provided in an answer to my own question below. My problem appears to be one of inexperience in working with ambient coordinates, ...
2
votes
1answer
85 views

Dimension of graphs (Differential Geometry)

I have a rather basic question about the dimension of a graph mapped by the exponential map. The problem is I can't really visualize it. It starts like that: Let $M$ be a smooth manifold of ...
-1
votes
0answers
31 views

Describing the shape of these level sets.

Given the function $f(x,y,z) = y$ defined on $T^{2} = \{(x,y,z) : (\sqrt{x^{2} + y^{2}} - R)^{2} + z^{2} = r^{2}\}$ - the torus with inner radius $r$ and outer radius $R$ satisfying $0 < r < R$ ...