(Stub) The study of differential geometry with notions of infinitesimal distance given by a Riemannian metric. This allows us to consider questions about the shape of a manifold by studying its curvature.

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The standard connection in a regular surface is symmetric

The concrete setting: Let $M\subset \mathbb{R}^3$ be a regular surface. A vector field $X$ in $M$ is a differentiable function $X: M\to \mathbb{R}^3$ such that $X(p)\in T_pM$. Here we are taking the ...
7
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3answers
288 views

Are the Ricci and Scalar curvatures the only “interesting” tensors coming from the Riemannian curvature tensor?

In studying Riemannian geometry, one learns about the Riemann curvature tensor $$Rm(X,Y,Z,W) = \langle \nabla_X\nabla_YZ -\nabla_Y\nabla_XZ - \nabla_{[X,Y]}Z, W\rangle$$ and its various symmetries. ...
3
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1answer
117 views

What are all isometry classes of the 2-sphere?

In topology, one learns how to classify the compact surfaces up to homeomorphism. And in fact, since "homeomorphic" and "diffeomorphic" coincide in dimension 2, we can classify the compact (smooth) ...
3
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1answer
133 views

What is the norm of the gradient of $f$ in normal coordinate?

Let $M$ be a Riemannian manifold and $f$ a smooth function on $M$. The Bochner formula proved in Schoen-Yau's book "lectures in Differential Geometry": (prop. 2.2) $$ \Delta |\nabla f|^2(p)=2\sum ...
3
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3answers
156 views

What does $ds^2$ mean and how does it specify a metric?

Let $H$ be the upper half-plane in $\mathbb{R}^2$. How does the following expression $$ds^2= \frac{dx^2+dy^2}{y^2}$$ specify a Riemannian metric on $H$? I don't understand what the expression means. ...
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24 views

Diagonable Kernels over a Riemannian Surface

This question is motivated by this paper. There, they develop a stippling method which requires a kernel to be diagonal. Meaning a symmetric bilinear function $K\colon \chi\times \chi\to \mathbb{R}$ ...
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1answer
152 views

Least distance on Riemannian Manifold

I've been doing some calculations of geodesics in different Riemannian Manifolds. More precisely I'm trying to compute, given two points on a Riemannian Manifold, the smallest distance between those ...
5
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1answer
142 views

Diffeomorphic riemannian manifolds and volume forms

Maybe the question will be stupid, but I'm a beginner in riemannian geometry... We have two riemannian manifolds $(M,g)$, $(\overline M,\overline g)$ and a diffeomorphism $F:M\rightarrow\overline M$ ...
2
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1answer
56 views

Bounding the injectivity radius from below.

Let $(M, g)$ be a finite-dimensional Riemannian manifold, and let $S \subseteq M$ be a compact set. I want to prove the following fact: There exists a number $\epsilon > 0$ such that the ...
3
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2answers
83 views

Critical paths for length cannot have kinks.

This problem is in Spivak's Differential Geometry (Ch.9 #37), and he gives a sketch of a proof which I have been unable to finish. So let's compute $\frac{dL(\overline{\alpha}(u))}{du}\mid_{u=0}$ ...
5
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1answer
640 views

How to understand the Second covariant derivative?

I am reading P. Li's lectures on Geometric analysis. On page 14, the author defines the second covariant derivative as follows: Let $f$ be a smooth function on $M$. $\omega_1, \cdots, \omega_n$ be a ...
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1answer
199 views

Connection between covariant derivative and basis vectors.

I read here, Curvilinear page 11, that $$\frac{\partial}{\partial x^i}e_j=\Gamma^k_{ij}e_k$$ where the $e_i$'s are basis vectors. There seems to be some connection, but when I calculate it, for ...
3
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0answers
127 views

What is the Weitzenböck formula for the $\bar\partial$-Laplacian?

It is well-known that the Weitzenböck formula for the real Laplacian is $$ \Delta |\nabla f|^2 =|\operatorname{Hess} f|^2 + \langle \nabla f, \nabla \Delta f\rangle + \operatorname{Ricci}(\nabla f, ...
3
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0answers
122 views

Length of closed geodesics in a compact and simply connected manifold X

I have a Riemannian manifold $(X,g)$ which is compact, simply connected and with sectional curvature upper bounded by $k>0$ everywhere. Let $p\in X$ be any point and $q\in Cut(p)$ the nearest cut ...
2
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1answer
298 views

control of the $C^{1}$ norm of a diffeomorphism

Let $\mathcal{E}$ be the set of smooth manifolds with boundary $E\subset \mathbb{R}^{3}$ which are perturbations of the unit ball whose volume $V$, diameter $d$ and area of the boundary $A$ satisfy: ...
6
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1answer
385 views

Prerequisites for studying Hodge theory and the Hodge conjecture

To what branch of mathematics does the Hodge conjecture belong? I'm aware that it's very advanced, but what kind of prerequisites would one need to understand those problems? Can you suggest some good ...
3
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2answers
112 views

Invariance of curvature under a conformal mapping

Let $\Omega_{1}, \Omega_{2} \subseteq \mathbb{C}$ be bounded domains. Let $\rho$ be a metric on $\Omega_2$ and $h: \Omega_1 \rightarrow \Omega_2$ a conformal mapping. Let $$h^*\rho(z) = ...
5
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1answer
205 views

showing zero curvature implies a line

How can I show that a given (not necessarily unit-speed) parametrization $\gamma(t)$ of a curve in $\mathbb{R}^3$ which exhibits zero curvature is a line ? What I know is that zero curvature means ...
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1answer
130 views

curvature of space curve

I am slightly confused by the following curve $\gamma(t) = (e^t,0,0)$ in $\mathbb{R}^3$. Its curvature, defined as $$ \kappa(t) = \frac{\|\dot \gamma(t) \times \ddot \gamma(t)\|}{\|\dot \gamma(t)\|^3} ...
12
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2answers
436 views

Computation of Laplace-Beltrami operator in a conformally equivalent metric

Could anyone tell me where I'm wrong with the following elementary calculation? Given a smooth Riemannian manifold $(M, g)$, I'd prove that if $\tilde{g}$ is conformally equivalent to $g$ (that is, ...
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0answers
96 views

Existence of Spin Group

"In mathematics the spin group Spin(n) 1[2] is the double cover of the special orthogonal group SO(n), such that there exists a short exact sequence of Lie groups As a Lie group Spin(n) therefore ...
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4answers
294 views

Is length adimensional when space is not flat?

Consider the two manifolds $\mathbb{R}^2$, equipped with the usual metric $g_{ij}=\delta_{ij}$, and $\mathbb{H}^2=\{(x, y)\,:\,y>0\}$, equipped with the hyperbolic metric $h_{ij}=\delta_{ij}/y^2$. ...
5
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1answer
430 views

good problem book in differential geometry

What are the books in Differential Geometry with good collection of problems. At present I am having John M.Lee's Riemannian Manifolds,Kobayashi Nomizu's Foundations of Differential Geometry. I ...
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2answers
124 views

Equivalence of intrinsic and extrinsic metrics of embedded manifolds.

Say a compact n-manifold $\mathcal{M}$ is embedded in $\mathbb{R}^m$, $m > n$. If $d_{\mathcal{M}}$ is the geodesic distance on $\mathcal{M}$, and $d$ the Euclidean distance in $\mathbb{R}^m$, ...
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134 views

Is $\bar{\partial}_E + \bar{\partial}_E^*$ a Dirac operator?

I have previously asked about Weitzenböck identities and received some great answers on MathOverflow. One question which has arisen from the post is the following: Let $E$ be a hermitian ...
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0answers
67 views

Solving to get free falling coordinate as function of arbitrary coordinate

From weinberg's gravitation, EQ : $3.2.11$ $$\frac{\partial^2 \zeta^\alpha}{\partial x^\mu \partial x^\nu} = \Gamma^\lambda _{\mu \nu}\frac{\partial\zeta^\alpha}{\partial x ^\lambda}$$ The solution ...
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1answer
325 views

Dolbeault Cohomology is invariant under homeomorphisms

If $X$ and $Y$ are two complex manifolds, which are homeomorphic but not necessarily diffeomorphic, must their Dolbeault cohomology groups be isomorphic? Here the Dolbeault cohomology groups ...
3
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1answer
224 views

Covariant derivative of a pushforward

Suppose that $\Phi_t$ is a the global flow associated with a vector field $X$ on a Riemannian manifold $M$ and that $Y$ is any other vector field. Suppose furthermore that $X$ is a Killing vector ...
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0answers
84 views

a problem of comparison geometry: Riemannian manifold with upper bounded sectional curvaturee

I have a question about Comparison Geometry: I have a Riemannian Manifold $(X,g)$, complete and simply connected, with sectional curvature upper bounded by a positive constant $k>0$, so I can ...
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0answers
161 views

Correct use of substitution rule for Integration on Riemannian manifolds

Let $(N,g_N)$ be a Riemannian manifold and let $\psi: M \rightarrow N$ be a a diffeomorphism. Now I know how the Riemannian metric on $M$ defined by the pull-back of the metric on $N$ looks like (this ...
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48 views

Holomorphic analogue of geodesics

Let $X$ be a complex manifold with a Hermitian metric. Is there a "complex" analogue of geodesics on $X$ which is of any interest? For example, is anything known about holomorphic maps $f : \mathbb C ...
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0answers
100 views

Volume of a ball intersected with a submanifold

I am given a smooth Riemannian submanifold of dimension $d$ embedded in $\mathbb{R}^D$ with condition number $1/\tau$ (a formal definition of condition number is on Page 3 of this paper ...
2
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1answer
118 views

notation question - vector field and function on manifold

So I'm trying to learn Riemannian geometry on my own... probably not a realistic goal! But anyway, for now I'm stuck on understanding part of this passage: A vector field $X$ on a $C^{\infty}$ ...
5
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2answers
163 views

Tensor Components

I would like to ask something On the Barrett Oneill's Semi-Riemann Geometry there is a definition of tensor component: Let $\xi=(x^1,\dots ,x^n)$ be a coordinate system on $\upsilon\subset M$. If $A ...
2
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1answer
85 views

Open sets of the tangent bundle in a Riemannian manifold

Let $M$ be a Riemannian manifold with a metric $g$ and $(U,\varphi)$ a chart around a point $p\in M$. By a Remark page 63 of Riemannian Geometry by M. Do Carmo, it seems that any open set ...
3
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1answer
241 views

Exponential map on manifolds and differential

I am trying to understand the proof of Theorem 3.7, page 72 of Riemannian Geometry by M. Do Carmo. For $M$ a Riemannian manifold and $(U,\varphi)$ a chart around a point $p\in M$, he (more or ...
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1answer
107 views

Holonomy group of quotient manifold

Let $(M,g_M)$ be a compact Riemannian manifold with holonomy group $Hol(M,g_M)$. Suppose that a finite group $G$ acts on $M$ freely and preserves the metric $g$. What can one say about the holonomy ...
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1answer
71 views

in Riemannian geometry, when is there an ambient space?

I am reading Kuhnel's Differential Geometry of Curves,Surfaces,Manifolds (2ed). On p.209, discussing tangent space of riemannian manifold, it says: ``since there is no ambient space, this notion has ...
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1answer
199 views

Isometry and harmonic forms

Let $M$ be a Riemannian manifold. Assume that a finite group $G$ acts on $M$ as isometry. How can one prove that $G$ takes harmonic forms to harmonic forms?
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2answers
356 views

do Carmo: Second Variation Formula

I have some trouble with the derivation of the second variation formula in do Carmo's famous "Riemannian Geometry" (p. 197f.). The proposition is the following: 2.8 Proposition Let ...
5
votes
1answer
261 views

A vector bundle admits a local covariantly constant section iff it is flat

Let $p:E\rightarrow M$ be a vector bundle over a manifold $M$ and let $\nabla$ be a connection on $E$. I am trying to show that $E$ admits a covariantly constant section $s$ in a neighborhood of each ...
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0answers
64 views

Convex polyhedral decomposition of spheres

Is there a decomposition of $S^2$ into $k$ (geodesically) convex polyhedra that are congruent to each other? What about $S^n$ for $n>1$? Remarks: A polyhedron is defined as an area enclosed by a ...
12
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2answers
360 views

metric in the Wasserstein space of gaussian measures

I am reading the paper "Wasserstein Geometry of Gaussian measures" by Asuka Takatsu (section 3 is of interest to me) and I have difficulties understanding how the metric is used. In particular, I am ...
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2answers
883 views

PDEs on manifold: what changes from Euclidean case?

I know some PDE theory for nice open domains in $\mathbb{R}^n.$ I want to know what the changes are when I switch to other domains like manifolds. For example, do things like Poincare's inequality ...
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2answers
115 views

terminology question - exponential map

The exponential map goes from the tangent space to the manifold, and the log map goes back. In reading, however, I get the impression that people use the "exponential map" as a term for the overall ...
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3answers
2k views

Riemannian/Ricci curvature for round n-sphere

What is the best way to see that the Ricci scalar curvature of $(S^n(r),g_{round})$ is a constant $n(n-1)/r^2$ ? I essentially only see this value stated in the literature, but no computation ...
7
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1answer
462 views

Directional derivative of vector field

I am trying to compute the directional derivative of a vector field $V$ along a direction $U$. Actually, my vector field is initially only defined on a curve $\gamma(t)$ in a Riemannian manifold $(M, ...
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1answer
394 views

Commutators, and Christoffel symbols in a non holonomic basis

I have a frame that varies along a curve $\gamma$ : the frame consists in the tangent vector of the curve plus some constant non orthogonal vectors. I need to compute ...
3
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1answer
147 views

Derivative of a parallel translation inside a metric

Let $M$ be a riemannian manifold with metric $g$ and a connection $\nabla$ on $M$. Let $X,Y$ two vector fields along a curve $\gamma$ on $M$. Let $$\tau_{t,s}:T_{\gamma(s)}M\to ...
5
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2answers
214 views

Is there an “opposite” of a geodesic?

If I understand correctly, a geodesic between two points $a$ and $b$ is the "most direct" path from $a$ to $b$. Geodesics on a plane are straight lines, geodesics on a sphere are great circle arcs. ...