Questions tagged [differential-geometry]

Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses on "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such structures, use (differential-topology) instead. Use (symplectic-geometry), (riemannian-geometry), (complex-geometry), or (lie-groups) when more appropriate.

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how we can compute $\hat{i}(\alpha,\beta)$ and ${i}(\alpha,\beta)$ for following curve?

in the Farb and Margalit: A primer on MCGs. on page 28 we have : There are two natural ways to count the number of intersection points between two simple closed curves in a surface: signed and ...
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Approximate a convex body with a domain with smooth boundaries?

Let $K$ be a convex body in $\mathbb R^n$ (i.e. a set that is compact and convex). Is it possible to construct a domain with smooth boundaries that contains $K$ and is close enough in the sense of ...
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how the angle between vectors changes in a topological equivalence

I'm studying dynamical systems and i have the next problem: I have two dynamical systems in continuous time, let's say $\dot{x} =X(x)$ and $\dot{y} =Y(y)$ where there is a topological equivalence (or ...
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Topological properties of shortest paths

Suppose we have a manifold $M$ equipped with a reasonable notion of arc length $l(\cdot)$ for smooth paths. I guess by "reasonable" I mean at least that $l(0)=0$, $l(a+b)=l(a)+l(b)$, and $l(...
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Proof that the shape operator is $\rm I^{-1} II$

I will omit the source of the problem to avoid the possibility of an outright answer to a book exercise. The problem is to prove that the matrix of the shape operator for basis vectors $r_u, r_v$ is $\...
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Do unique horizontal lifts of path homotopic paths through a point are always path homotopic?

Let $\pi:E \rightarrow M$ be a principal $G$-bundle for a Lie group $G$. Let $\omega$ be a connection on the principal bundle. It is a well known fact, that if a path $\gamma:[0,1] \rightarrow M$ is ...
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What is the "gauge field" on the base space in a gauge field theory?

Suppose we have a principal $G$-bundle $P\xrightarrow{\pi} M$, and we want to consider a classical gauge field theory (with a field Lagrangian) on $M$ for this bundle, in the physics sense. In the ...
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Calculating the product of the Riemannian manifolds and the Riemann curvature [closed]

I am a physics Master student currently taking a course on Riemannian geometry. In the course we are supposed to solve problem 7-4 out of Lee's Book Introduction to Riemannian Manifolds. The Problem ...
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Does tautological line bundle $\mathcal{O}(-1)$ over complex projective space have zero section?

In Complex Geometry written by Huybrechts, it defines the tautological line bundle $\mathcal{O}(-1)$ as followed $$ \mathcal{O}(-1):=\{(\ell,z)\in \mathbb{C}P^n \times \mathbb{C}^{n+1}: z\in \ell\} $$ ...
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Weak differentiability of SDEs on manifolds

For a SDE on Euclidean spaces, it is a well known result that solutions of such SDE is smooth (with respect to Malliavin derivative on the path space of the Euclidean space) when the coefficients are ...
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Is there a distribution not expressible as a kernel of a list of 1-forms?

Given a smooth manifold $X$ and a smooth distribution/vector-subbundle $E\subset TX$ on $X$ of codimension $k$, is it possible that there do not exist $\omega_1,\ldots,\omega_k\in \Omega^1(X)$ such ...
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Does the Differential Topology/Geometry frameworks being able to model solutions to diff. eqs. that are Non-Smooth?

I don't have much knowledge about Differential Topology neither Differential Geometry, but working on this another question about solutions to differential equations, and someone recommend me to ...
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Factorial term when evaluating the exterior derivative through the connection definition

I found perhaps a hint of what the relation is on page-316 of Penrose's Road to reality. For $ p- $ form $\alpha$ with index epxression, $\alpha_{b...d}$ , and a torsion free connection $\nabla_a$ $$(...
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Pulling back Lie-valued forms to get curvature?

According to Lemma 1 in https://en.wikipedia.org/wiki/Chern%E2%80%93Weil_homomorphism#Definition_of_the_homomorphism , if $\Omega$ is the curvature of a connection on a principal $G$-bundle $P\...
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What is the precise definition of the Darboux tangent to a surface? [closed]

What is the definition of Darboux tangents of a surfaces? The book "Affine Differential Geometry" mentions it, but it does not give a precise definition.
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Showing that a curve from vertex $A$ of $\triangle ABC$ to side $BC$ intersects a curve from $C$ to side $AB$ at least once

Suppose that we have a triangle $\triangle ABC$, and two continuous curves inside it: one starts at the vertex A and ends on the side $\overline{BC}$, and the other one starts at the vertex $C$ and ...
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Reconstruct a Euclidean motion from its action on a curve

I have an issue with an exercise regarding the reconstruction of a euclidean motion. Let $\gamma: [0,L] \to \mathbb{R}^2$ be an arclength-parametrized closed curve and $A: \mathbb{R}^2 \to \mathbb{R}^...
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Pullback of a constant coefficient form

Let $\omega=\sum_{I}C_Idx_I$, where $I=(i_1,\dots,i_n)$ is a multi-index and $C_I$ constants, be an $n$-form in $\mathbb{R}^m$, with constant coefficients. Here $dx_I$ means $dx_{i_1}\wedge\dots\wedge ...
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What does it mean when a chern class of a vector bundle is postive(resp. negative)?

Recently i was studying line bundles on $\mathbb{C}P^1$. Here is my confusion: for any holomorphic map $f:\mathbb{C}P^1 \to M$, where $(M,E,\nabla)$ is a $r$-rank holomorphic vector bundle with a ...
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Regarding the decomposition of a tensor as an antisymmetric and symmetric part

I have seen for two index tensor that the sum of symmetric and anti symmetric part is the total tensor again.. but is it true for higher index tensor? I thought it was but then I saw this diagram in ...
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2 votes
1 answer
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Missing minus sign in pullback calculation of a $1$ form on $\mathbb{C}$

Consider $S^2$ with a coordinate chart given by the stereographic projection through the north pole. We identify $\mathbb{C}$ with $\mathbb{R}^2$, then for a point $(\theta,\phi)\in S^2$, the ...
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4 votes
1 answer
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Splitting the Tangent Bundle of a Vector Bundle along the Zero Section

Good evening everyone, I have a small question: Assume we have a vector bundle $E = \bigcup\limits_{x\in M} E_x$ over a manifold $M$. I now want to show the following well-known equation: $$TE_M \cong ...
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5 votes
3 answers
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How to reconcile two approaches to the Gaussian curvature

Gaussian curvature can be found as the ratio of determinants of the second to the first fundamental forms (from here): $$K =\frac{\det \mathrm {II}}{\det \mathrm I}=\det \left(\mathrm {I}^{-1}\rm {II}\...
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What is the formula for the orbital velocity of the Earth from x, y, z coordinates of ephemerides at set time intervals? [closed]

For example for step size 10080 minutes x y z v 1721057.5 B.C. 0001-Jan-01 -5.83E-01 7.93E-01 3.65E-03 1721064.5 B.C. 0001-Jan-08 -6.78E-01 7.16E-...
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Mathematically, why does moving the vanishing points correspond to rotations of a 3D figure in space?

I was watching this video on how vanishing point changes as object rotates, in it, it is shown that as we move the vanishing points of the family of extended cube sides , the new configuration of ...
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Isomorphism of pseudo-orthogonal group $O(p,q)$ and $O(p', q')$ for $p,q, p',q' \in N$ [closed]

Let $O(p,q)$ be the pseudo-orthogonal group or indefinite orthogonal group of signature $(p,q)$ as described in https://en.wikipedia.org/wiki/Indefinite_orthogonal_group. Suppose $O(p,q)$ is ...
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The connection between a Metric Tensor and a Metric in Topology

Does the Metric tensor define a metric on the topology of the manifold/space it's on, and/or does the metric on the topology of the space define a metric tensor, if you go through all the trouble of ...
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Why the poincaré theorem is not possible in $\mathbb R^n$?

Hello I am reading about the Poincaré-Bendixson theorem in the plane and then in compact two-dimensional manifolds. But I have some doubts that I would like you to help me clarify: I know the example ...
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Is this geometric surface (flat torus) in $\mathbb R^3$ or not?

I'm reading Barrett O'neill's Elementary Differential Geometry, and this example has me confused: and the parametrization from Example 2.5 is: lastly, Theorem 3.5 is: On every compact surface $M \...
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1 vote
1 answer
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A square on the equator of a sphere is a critical point of the electrostatic potential

$\newcommand{\S}{\mathbb{S}^2}$ This is a self-answered question. I learned something from spelling out the details, and I hope this could be interesting to others. I would welcome alternative ...
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3 votes
2 answers
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Is this the correct understanding of how a geometric surface works?

I'm reading "Elementary Differential Geometry" by Barrett O'Neill. Most of the book is spent looking at surfaces in $\mathbb R^3$, but eventually he introduces the "abstract surface&...
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What is $\partial_t g^{ij}(t)$?

This is probably a dumb question; so far, I have been using $g^{ij}g_{ij} = n$ as granted, but I am probably doing something wrong here. Suppose that $g_{ij}(t)$ is a time-evolving metric. Let $h_{ij} ...
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Proof that the covariant derivative is orthogonal to the tangent plane

Pavel Grinfeld in here writes: $$\vec S^\gamma \nabla_\alpha \vec S_\beta= \vec S^\gamma \cdot \frac{\partial \vec S_\beta}{\partial S^\alpha}=\Gamma^\omega_{\alpha \beta} \vec S_\omega\cdot \vec S^\...
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Straight line in projective plane is plane in vector space (Road to Reality , page-343)

How do we construct an n-dimensional projective space $P^n$? The most immediate way is to take an $(n + 1)$-dimensional vector space $V^{n+1}$, and regard our space $P^n$ as the space of the 1-...
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1 vote
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Gauss Bonnet for locally outermost closed minimal surface in Hawkings Theorem

So I'm trying to understand the proof for the following theorem from Lan-Hsuan Huang: "Trapped Surfaces, Topology of Black Holes, and the Positive Mass Theorem": Any orientable locally ...
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4 votes
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Are there tensor structures other than a metric which could be defined on a manifold which imply a connection through compatibility criterion?

If we say our connection is torsion free, then the metric compatibility condition completely determines it. While this is geometrically intuitive way to do it, are there other interesting tensor ...
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4 votes
1 answer
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Can mathematics distinguish left and right?

Imagine, a mathematician from another galaxy lands on the earth. Is there a way we can explain to him what is "counterclockwise" without showing him a picture? Things like Green's formula, ...
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Is a planar square on the equator a locally energy minimizing configuration of electrons on $\mathbb{S}^2$?

$\newcommand{\S}{\mathbb{S}^2}$Let$$M=\{(x_1,x_2,x_3,x_4) \in \mathbb{S}^2 \times \mathbb{S}^2 \times \mathbb{S}^2 \times \mathbb{S}^2 \, |\,\, \text{ all the } x_i \, \text{ are distinct}\} $$ Let $...
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Are there some vector calculus identities that you can't prove using differential forms? [duplicate]

There are some vector calculus identites involving operators like $(v \times \nabla)$ , $ (v \cdot \nabla)$ etc (concrete example). To my knowledge, there is no similar equivalent such in Forms, does ...
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3 votes
1 answer
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Help proving statement in Lee's Introduction to Riemannian Manifolds about smooth curves into manifolds with nonempty boundary.

The statement appears on page 33 of the second edition of Professor Lee's Introduction to Riemannian Manifolds. It is in the section on Lengths and Distances in Riemannian manifolds, but I think the ...
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2 votes
1 answer
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Regarding a proof that $T(M\times N) \cong TM\times TN$

I want to ask about the answer in this link: Tangent Bundle of Product Manifold How is the identification $T_{(x,y)}(M\times N)=T_xM\oplus T_yN$ used in (*) ? And in writing $T(M\times N)$ and $TM\...
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Which one should I start studying - Lie Groups/Algebras or Functional Analysis or Differential Geometry? [closed]

I major in Physics(an excuse for sloppy math practices) and wish to go deeper in terms of understanding the concepts and improving mathematical rigorousness. I wish to study Algebraic Topology, ...
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2 votes
1 answer
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definition of lie bracket of linear connection and a $\operatorname{End}(E)$-valued $k$-form

I'm currently studying LECTURES ON CHERN-WElL THEORY, in which it defines the trace function on a vector bundle $(M,E)$ by setting $$ \operatorname{tr}(\omega\otimes A) = (\operatorname{tr}A)\omega ,\...
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1 vote
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Extrinsic curvature using different coordinates

I have some clarifications about the calculation of the normal vector and extrinsic curvature of a cylinder on pp. 25-26 (gr-qc/0703035), The Euclidean metric in cylindrical coordinates $(r,\phi,z)$ ...
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4 votes
1 answer
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Calculation on Riemannian manifolds

I am learning the variational calculation of Yang Mills functional, but I can't understand 2 steps in the following calculation: Given a variation of the connection $A$ in local coordinates: $A\to A+\...
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3 votes
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Pullback of a density by the exponential map

In this lectures notes Geometric wave equation by Christian Bär at page 17 he has Definition 1.2.27. Let $\Omega$ be a starshaped with respect to $x$. We define the smooth positive function $\mu_{x}: ...
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3 votes
3 answers
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Can a manifold be reconstructed from its charts?

I'm learning special relativity and I am having a confusion on this mathematical point. Whenever any sort of motion or non motion happens in the world, it can only be perceived by the scientist in a ...
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In the proof of corollary associated to the closed subgroup theorem

I'am reading the John M. Lee, Introduction to Smooth manifolds and I have a question. Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$. And let $\operatorname{exp} : \mathfrak{g} \to G$ be the ...
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9 votes
1 answer
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What is an instanton? (On a complex surface or a differentiable 4-manifold )

The question is as in the title. I have browsed online (Wikipedia, etc) and while they do give me the definition, it gets a bit too much physics-y for me. Therefore I would appreciate it if someone ...
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3 votes
1 answer
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Is the map $T_X |_S (p) := \exp{X(p)}$ a diffeomorphism onto its image?

Preliminaries The exponential map $\exp : TM \rightarrow M$ is defined by $\exp{(v)} = \gamma_v (1) $ where $\gamma_v$ denotes the geodesic starting at $p \in M$ and initial velocity $v \in T_p M$. ...
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