A branch of differential geometry dealing with Riemannian manifolds. *Riemannian manifolds* are smooth manifolds with an inner product smoothly attached to the tangent space of each point. Usually, Riemannian geometry focuses on the notions of distance, curvature, and shape. Consider using this tag ...

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Existence of minimizing geodesic in each fixed-end-point homotopy class in a complete manifold?

This is intuitively clear, but I cannot solve this homework problem: 1) Let $(M,g)$ be a complete Riemannian manifold, let $c:[0,1]\to M$ be a continuous curve in $M$ such that $c(0)=p, c(1)=q$. Then ...
12
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1answer
453 views

Do eigenfunctions of elliptic operator form basis of $H^k(M)$?

We know that the eigenfunctions of the Laplacian on a compact manifold $M$ form a countable basis of $H^1(M)$. If $L$ is a $2k$-order elliptic operator, do the eigenfunctions of $L$ form a basis for ...
4
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53 views

A Simons' type inequality

I have a problem with the inequality (5) in the article 'Estimates for stable minimal surfaces in three dimensional manifolds' of R.Schoen. As the author suggests this inequality comes from 'well ...
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0answers
79 views

Normal Bundle of Twistor lines

I am reading the paper "hyperkaehler metrics and supersymmetry" by Hitchin etc.. Here is the link: ...
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1answer
133 views

Angle preserving transformation

I've been working on a problem where I need to know the angle between the tangent vectors of two curves at their intersection point in a flat torus... Then I thought: Consider two geodesics ...
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1answer
74 views

Singularities of flat manifolds

Can a flat manifold have infinitely many singularities? Let me explain: I am working on riemannian geometry applied to thermodynamics. I am analyzing closed simple systems whose state of equilibrium ...
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241 views

Which coefficients of the characteristic polynomial of the shape operator are isometric invariants?

Let $M^n \subset \mathbb{R}^{n+1}$ be an isometrically immersed Riemannian hypersurface. The shape operator $s$ is the $(1,1)$ tensor field characterized by $$\langle X, sY \rangle = \langle ...
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1answer
532 views

Isometry and geodesic

Let $F: M \rightarrow N$ an isometry and $M,N$ two riemannian manifold. How can I prove that the set of fixed points of F isometry (among riemannian manifold) is a geodesic? In general is it a curve?
3
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578 views

Geodesics on a 2-sphere

I've been doing some work where I need to find the geodesics in a given Riemannian Manifold. Let's take the example of the two sphere, for simplicity, with unitary radius. The distance between two ...
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368 views

How to derive covariant derivative and Lie derivative of tensors

1) As title says, how does one derive the following equation for covariant derivate of tensor: $A^{\alpha}{}_{;\beta} = A^{\alpha}{}_{,\beta} + \Gamma^{\alpha} {}_{\gamma\beta}A^\gamma$ where ...
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294 views

Intuition for Fisher information metric

In statistical maniolds $S=\{p_\theta\}$,$\theta=(\theta_1,\dots,\theta_n)$, the Riemaanian metric usually defined is the Fisher information metric $$g_{ij}(\partial_i,\partial_j)=\int \partial_i(\log ...
6
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1answer
184 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 ...
3
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1answer
236 views

Maximum principle for minimal hypersurfaces

The well known local version of the maximum principle for minimal hypersurfaces asserts that if two minimal hypersurfaces $ M_1 $ and $ M_2 $ of $ R^n $ has a common point $ x_0 \in M_1 \cap M_2 $ ...
3
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1answer
226 views

A question about laplacian of the second fundamental form

Let $ f:M \rightarrow N $ be an immersed oriented hypersurface, $ e_{1}, \ldots e_{n},e_{n+1} $ be an orthonormal frame of $ N $ such that $ e_{1} \ldots e_{n} $ is an orthonormal frame of $ M $. Let ...
6
votes
3answers
514 views

Elliptic estimates on compact manifolds

Hey where may I find elliptic estimates for PDEs on compact (no boundary) Riemannian manifolds? I want a source/paper/book where I can cite it. For example, for $L$ a linear elliptic operator, (eg. ...
2
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1answer
47 views

Schoen Estimates (part 3)

I'm referring to the article 'Estimates for stable minimal surfaces in three dimensional manifolds' of Richard Schoen In the first paragraph of the proof of theorem 2 the author seems to assert that ...
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0answers
153 views

Riemannian metric for euclidean geometry

I am a beginner to Riemannian geometry. Following is my question. In the Euclidean space, say $\mathbb{R}^3$, let us consider a plane, for simplicity, say one passing through the origin, ...
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1answer
103 views

Regular compact domains of a Riemannian manifolds

In a Riemannian manifold $ M $ a regular compact domain $ D $ is a compact subset of $ M $ with non empty interior and such that for every $ p \in \partial D $ there exists $ \left(U,\varphi\right) $ ...
2
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1answer
62 views

Existence of vector fields

Does there exists two vector fields $X$ and $Y$ on $\mathbb R^2$ such that the following are satisfied? $X(0)= Y(0)= 0$, where $0\in \mathbb R^2$ and for others points $q\in \mathbb R^2$, we ...
3
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0answers
64 views

Schoen curvature estimates (part 2)

I'm referring to the article ''Estimates for stable minimal surfaces in three dimensional manifolds'' by Richard Schoen. I have a question about the proof of theorem 1. In the last step of the proof ...
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1answer
60 views

A question about an estimate

Let $ f:M \rightarrow N $ be a minimal immersion where $ M $ is a compact two dimensional manifold and $ N $ is a three dimensional manifold. Let $ |A|^2 $ be the square of its second fundamental ...
2
votes
1answer
320 views

Thoughts about sectional curvature

I'm currently trying to understand the sectional curvature of riemannian manifolds and I don't know if I'm thinking correctly. So, say we have a riemannian manifold $(M,g)$ with constant sectional ...
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1answer
244 views

why is the spectrum of the schrödinger operator discrete?

let (M,g) be a compact riemannian manifold. Then the spectrum of the Schrödinger opartor $H=-\Delta +V$ with bounded potential V acting on $L^2(M)$ consists of discrete Eigenvalues ...
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67 views

$|\Delta f|^2$ in local coordinates

The Laplace-Beltrami in local coordinates (for hypersurfaces in my case) is $$\Delta f = \frac{1}{\sqrt {|g|}} \partial_i \left(\sqrt{|g|} g^{ij} \partial_j f \right)$$ Is there a nice formula for its ...
3
votes
2answers
881 views

Divergence theorem on Hyperbolic space

Given a vector field, say $F$, defined on a manifold $U$, the divergence theorem states that: $$\int_U\nabla \cdot F dV=\int_{\partial U} F d \Sigma .$$ Well if the manifold is $\mathbb R ^n$ and $F$ ...
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2answers
129 views

Constant Riemannian Metric

Let $M=\mathbb R^n$ and define for each $x\in \mathbb R^n$, define $$\langle v,w\rangle_x= \langle v,w\rangle_0$$ where $v,w\in T_x\mathbb R^n\equiv \mathbb R^n\equiv T_0\mathbb R^n$. Hence we see ...
3
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1answer
299 views

Is there a non-variational derivation of Snell's law from Fermat's principle?

Every proof I've seen of Snell's law from Fermat's principle uses some sort of variational argument, mostly involving variational calculus. Niven's wonderful book, Maxima and Minima Without Calculus ...
4
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1answer
176 views

Manifold without conjugate points and positive curvature

I'm looking for an example of a complete riemannian manifold with sectional positive curvature and without conjugate points. I've tried the projective space, but the identfication used to construct it ...
5
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1answer
137 views

Riemannian metric. Help with notation.

I was just reading about the hyperbolic space (upper-half plane model) and i'm getting kind of confused about the notation for the Riemannian metric. The half-plane is defined as $$ H = \{(x,y) \in ...
4
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2answers
952 views

Riemannian Geometry book to complement General Relativity course?

What would be a good Riemannian Geometry (or Differential Geometry) book that would go well with a General Relativity class (offered by a physics department)? I'm in one right now, but I'd like a pure ...
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0answers
85 views

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 ...
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355 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. ...
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1answer
136 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
148 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 ...
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votes
3answers
169 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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25 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
173 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
156 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
58 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
votes
2answers
87 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
758 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
254 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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1answer
155 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, ...
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146 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 ...
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1answer
319 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: ...
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1answer
461 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
130 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
233 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
140 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} ...
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2answers
545 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, ...