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 ...

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show $\omega$ is exact form

Let X be the region $\mathbb{R^3}-(0,0,0)$ and f(x,y,z) is $C^\infty$ function on X. Also $\omega$ is 1-form $f(x,y,z)(xdx+ydy+zdz)$. if $f$ can be expressed in the form $f(x,y,z)=h(r)$ when ...
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Sign of Laplacian in a N-Dimensional Space Under Other Specific Conditions

Consider that $f: \mathbb{R}^N \to \mathbb{R} $ is infinite times differentiable and is smooth over the entire domain. Take $N$ to be large. Also at point $\mathbf{x_0}$: $$ ...
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Finding an isometry that maps one circle to another.

I have a problem goes as follows: Consider the unit speed curve $$\boldsymbol{r}(s)=\left(\frac{4}{5}\cos(s),1-\sin(s),-\frac{3}{5}\cos(s)\right).$$ Find an isometry $f$ such that ...
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38 views

A Problem from Docarmo's Differential Geometry

The following is a (may be simple) problem from Docarmo's Differential Geometry. Let $\alpha\colon (a,b)\rightarrow \mathbb{R}^3$ be a parametrized curve which do not pass through origin. If ...
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43 views

Use contraction mapping theorem to prove

Help! I am taking a math course, and I just can't figure out this proof: Let $\alpha,\beta\in R^n$, $a\in R$, and $A$ be an $n\times n$ nonsingular matrix. Use contraction mapping theorem to prove ...
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46 views

Hessian is proportional to the metric everywhere

Let $(\Omega^{n+1},g)$ be a compact Riemannian manifold with smooth boundary. Let $f\in C^{\infty}(\bar{\Omega})$ satisfies $\operatorname{Hess}f=\frac{1}{n+1}g.$ Suppose the minimum of $f$ occures at ...
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65 views

Differential Geometry Intuition Question

Apologies if I get the notation wrong. Still learning this stuff. Suppose I have a 2 dimensional Riemannian manifold $\mathcal{M}$ that is covered by a single chart: $\phi: \mathcal{M} \rightarrow ...
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27 views

Exponential of a power of the differential operator

In relation to this question: Exponential of a polynomial of the differential operator Is there an expression for $\exp(aD^n)f(x)$ similar to $\exp(aD)f(x)=f(x+a)$?
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What does this operator $\odot$ mean

I read this about the second fundamental form in Wikipedia and I’ve no idea what does $\odot$ mean? Does anybody know? $$II=-dN\cdot dP=\omega^3_1\odot\omega^1+\omega^3_2\odot\omega^2$$
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defining smooth functions on smooth manifolds

The standard approach to defining smooth functions $f:M\to\mathbb{R}$ on a topological manifold $M$ equipped with a smooth structure (i.e., a maximal smooth atlas) $\mathcal{A}$ is the following. Say ...
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Problem about Ricci Flow

On page 12 of "Lectures On Ricci Flow" by Peter Topping is written: In two dimensions, we know that the Ricci curvature can be written in terms of the Gauss curvature $K$ as $Ric(g) = Kg$. Working ...
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A question on relating $N$-Sphere with a $(N-1)$-cell in $\mathbb{R}^{N-1}$

Let there be a $N$-Sphere in $\mathbb{R}^N$. Every point in it is a unit vector in $\mathbb{R}^N$. Every real valued function $f$ defined on this sphere accepts a unit vector $\hat{a}\in\mathbb{R}^N$ ...
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map between classifying spaces induced by group homomorphism

Let $\Sigma_n$ be the permutation group of order $n$. Then the regular representation of $\Sigma_n$ gives an injective homomorphism $f:\Sigma_n\to O(n)$. Why $f$ induces a map between their ...
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Calabi-Yau manifolds and immersion in real space [closed]

I'm reading some papers how to test extra dimensions in LHC experiments and they suggests CY manifolds as starting point. Is it possible that accelerator itself is made in higher-dimensional geometry ...
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35 views

Minkowski space is locally Euclidean?

The Minkowski spacetime $\mathbb{R}^{1,3}$ is said to be a manifold (isomorphic to $SO^{1,3}$. But according to the definition of a manifold it should be locally euclidean. However, this seems to be ...
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Invariants of shape operators?

Let $S:V\longrightarrow V$ be a Linear Transformation, then the Characteristic polynomial of $S$ and therefore its Coefficient are invariant. Except the first and the last Coefficient that we know ...
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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 ...
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Why is the Leibniz rule a sufficient ingredient in the construction of the tangent space?

This is a very soft question, but I am wondering if anyone can shed light on why it is that the product rule (and linearity) provide exactly the right requirement for the space of derivations to be ...
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Yarn-like functions

When wrapping yarn around a ball you cannot make sharp turns or the yarn will fall off. If we think of the yarn as a curve on the surface of the sphere, we would say it must have curvature less than ...
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3answers
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Books on differential geometry in the cases $n=2$ and $n=3$

I'm interested in learning the differential geometry of standard, "physical" space, that is $\mathbb R^2$ and $\mathbb R^3$. The sort of problems that were studied in the 18th and 19th century... ...
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31 views

What 's conditions on open set related to connected neighborhood of boundary

I have a question: Suppose $D$ is an open set in $\mathbb{R^n}$ and topological boundary $bD$ is an embedded submanifold of $\mathbb{R^n}$. For each $p\in bD$, we want to have an open neighborhood ...
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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 ...
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94 views

The relation between geodesics and distances on a Riemannian manifold

My question is about computing the distance between two points in a Riemannian manifold. Suppose that $(M,g)$ is compact so that it is geodesically complete and geodesically convex. Let ...
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3answers
292 views

Parameter Curves are Geodesics

So let's suppose we have a surface $M$ that is embedded in $\mathbb{R}^3$ with an orthogonal parametrization. Further, assume that the parameter curves (i.e., $X(u$0$, v)$ and $X(u, v$0$)$ ) are ...
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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 ...
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Equivalence between Kähler condition and $\partial_kg_{i\bar j} = \partial_ ig_{k\bar j}$

Let $(M,\omega)$ be a Kähler manifold. Why is the Kähler condition $$d \omega = 0$$ equivalent to $$\partial_kg_{i\bar j} = \partial_ ig_{k\bar j}$$ for all $i, j, k$? I am looking for a reference.
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Trying to understand “derivative or Jacobian of smooth map”

From some lecture notes I am trying to puzzle through .... "... the derivative or Jacobian of a smooth map $f: \mathbb{R}^m \rightarrow \mathbb{R}^n$ at a point $x$ is a linear map $Df: \mathbb{R}^m ...
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Complex and Kähler-manifolds

I was woundering if anyone knows any good references about Kähler and complex manifolds? I'm studying supergravity theories and for the simpelest N=1 supergravity we'll get these. Now in the ...
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Differentiable manifolds $\mathscr C^k$ vs. $\mathscr C^\infty$

I noticed that there exists a (in some sense) better definition of the tangent space via the dual of a certain quotient algebra which is easier to work with in some cases. This however only works for ...
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$C^{k}$-manifolds: how and why?

First of all, I have a specific question. Suppose $M$ is an $m$-dimensional $C^k$-manifold, for $1 \leq k < \infty$. Is the tangent space to a point defined as the space of $C^k$ derivations on the ...
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40 views

Hard Lefschetz Thereom and the Cohomology of Flag Varieties

Let $G$ be a compact connected connected Lie group $G$, and $B$ a Borel subgroup containing a maximal torus. Moreover, let $F = G/B$ be the associated flag manifold. Now $F$ is a compact Kahler ...
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exercise on surfaces and geodesics

Maybe someone can verify my answers. The problem is as follows: Consider a line $l$ in $\mathbb{R}^3$ and rotate it around the $Oz$ axis. Denote by $A$ the set of all such obtained surfaces. ...
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Proof that this surface is of revolution

I have a surface with parametric equation $$\mathbf{x}(u,v)=(u\cos(v),u\sin(v),u^2),$$ $u$ is any real number, $v$ is between $0$ and $2\pi$. I don't know how to show that this is surface of ...
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If a plane intersects a regular surface at exactly one point, then it is the tangent plane

Question Let a regular surface, $S$, intersect a plane, $P$, at only one point, $p_0 = (x_0, y_0, z_0)$ in $\mathbb{R}^3$. Show that the plane coincides with the tangent plane to the surface at ...
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Definition of a lipschitz 1-form on a manifold

What is the definition of a Lipschitz-regular 1-form on a riemannian manifold?
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28 views

In uniform circular motion in R^2, is acceleration in the normal bundle?

In physics we learn that accleration is a vector quantity parallel to the radius and orthogonal to the velocity. With the embedding $\mathbb{S}^1 \hookrightarrow \mathbb{R}^2$ and the induced ...
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Invariant in geodesic

What in general is invariant in geodesic in terms of parameters $u$ and $v$ ( or functions on which they depend) and their derivatives in integrated form? For a surface of revolution, Clairaut's ...
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Restriction of a differentiable map $R^3\rightarrow R^3$ to a regular surface is also differentiable.

This is again an excercise from Do Carmo's book. Prove: if $f:R^3 \rightarrow R^3$ is a linear map and $S \subset R^3$ is a regular surface invariant under $L,$ i.e, $L(S)\subset S$, then the ...
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Meridians of surfaces of revolutions

First off, I know there is another question asking the same thing, but that one was concerning where to start, whereas for this one, I'm almost complete, but I can't get something at the end to ...
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3answers
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Showing that a Unit Speed Curve is a Circle.

In my recent differential geometry tutorial, we were given the question: Given the unit speed curve, $$\boldsymbol{r}(s)=\left(\frac{4}{5}\cos(s),1-\sin(s),-\frac{3}{5}\cos(s)\right)$$ show that ...
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48 views

Geodesic radius of curvature

I am trying to compute the geodesic (or tangent) radius of curvature of the geodesic circle by using the below formula. $\frac{1}{\rho_c}=\frac{\partial G/\partial S}{2\sqrt{E} G}$ where $s$ is the ...
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differential form identity and permutations

If $t^1,...,t^k$ are the coordinates of a k-cube. Then apparently $$dt^{\sigma(1)} \wedge \ldots \wedge dt^{\sigma(k)}= (\operatorname{sgn} (\sigma)) dt^1 \wedge dt^k $$ I cannot see how this ...
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How “far” a differential form is from an exterior product

Consider two differential manifolds $X$ and $Y$. Consider now a differential form (of any order) $\omega$ on $X\times Y$. The easiest example is taking $\omega=\xi\wedge\eta$, where $\xi$ is a ...
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Liouville form on the cotangent bundle

a) Let $(U,\phi) = (U,x^1,\dots,x^n)$ be a chart on a manifold $M$, and let $(\pi^{-1}U,\tilde {\phi}) =(\pi^{-1}U,\tilde {x^1},\dots,\tilde {x^n},c_1,\dots,c_n)$ be the induced chart on the ...
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Start of the proof of the Poincaré Lemma

Let $\hat{\Phi}:U \rightarrow U$ be a one-parameter family of diffeomorphisms defined for $\ 0 < t \leq 1$. Let $\beta \in \Omega^k(U) $ be a closed k-form. Suppose that $\hat{\Phi}^{*}_1 \beta = ...
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The differential $\text d F_p$ is injective iff the pullback $F_p^*$ is surjective.

I'm trying to prove the following claim: Let $F\colon M \to N$ be a differentiable application beetween $C^\infty$ manifolds. Then the differential $\text dF_p\colon T_p M \to T_{F(p)}N$ is ...
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contraction identity on $k$-forms

$i_\mathbb{X} \omega $ is the contraction of $\omega$ with respect to $\mathbb{X}$. In my notes it is stated that $i_\hat{\mathbb{X}} dx = dx(\hat{\mathbb{X_t}})$. I cannot see how this fits the ...
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General expression of smooth sections of tensor bundles.

On page 317 of John Lee's Smooth Manifolds it's said that if $(x^i)$ are local coordinates on a smooth manifold $M$, then sections of the tensor bundle $T^kT^*M=\bigsqcup_{p\in M}T^k(T^*_pM)$ over a ...
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Book on advanced Hodge Theory

I'm looking for a book on advanced real Hodge Theory. I finished working through Frank Warner's Foundations of Differentiable Manifolds and Lie Groups, which ends with the Hodge Decomposition,the ...
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Is Hodge star operation can be understood as contraction after tensor product of a $p$-form with the volume element?

By defintion, the Hodge star of a $p$-form $\omega_{a_1\cdots a_p}$ on a $n$-dimensional manifold is given by $*\omega_{b_1\cdots b_{n-p}}=\frac{1}{p!}\omega^{a_1\cdots a_p}\epsilon_{a_1\cdots ...