(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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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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geometrical consequences of nonpositive or negative Ricci curvature.

well,that's pretty much the question. I'd like to know if somebody could point me out if there's any geometrical implications following an upper bound on the Ricci curvature on a riemannian manifold. ...
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32 views

Comparing PDE solutions for different Riemannian metrics

I'm looking for the approach to compare PDE solutions on the Remannian manifolds when those solutions are obtained under two different metrics. To be more specific, suppose we have two Riemannian ...
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41 views

The invariance of the Ricci tensor under diffeomorphisms and its non-ellipticity.

Consider $(M,g)$ a compact Riemannian manifold. When viewed as a second order (non-linear) differential operator $$ \text{Ric} : C^{\infty}(\text{Sym}^2_+T^*M) \to C^{\infty}(\text{Sym}^2T^*M), $$ the ...
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Ovals of constant $ k_g$ on constant $K$ surfaces

Prove that: Constant geodesic curvature lines on constant Gauss curvature surfaces are closed Ovals/Loops. Find perimeter/length of this Oval/Loop in terms of $ k_g$ and $K$ I believe the proof ...
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35 views

$\Delta e^i =0$ where $e_i$ is geodesic.

Let $e_i$ be a geodesic coordinate vector field and $e^i$ be its coframe. Then $$\Delta e^i =0$$ This is right ? If so how can we prove ? $$\Delta e^i (e_j)=\nabla_k \nabla_k e^i(e_j) = e_k( ...
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Manifold characteristics in terms of Riemannian metric

I wonder what characteristics of Riemannian manifold can be expressed in terms of metric? Are there any results similar to Gauss–Bonnet theorem? Does the Riemannian metric give any information about ...
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80 views

Why are Euclidean and hyperbolic lengths proportional to first order?

In his book “Three-Dimensional Geometry and Topology”, Thurston constructs a Riemannian metric for the Poincare disk model and begins as follows. He says that, given any (hyperbolic) line segment $s$ ...
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48 views

surface in $R^3$ that has $ds^2 = du^2/v^2 + dv^2/v^2$

For a 2D surface, if we have the first fundamental form of $$ ds^2 = du^2/v^2 + dv^2/v^2$$, can we integrate it out to get the parameter form of the surface embedded in $R^3$? I tried something like ...
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Weakening Sobolev coefficient in an elliptic estimate.

One of the classical estimates for Sobolev norms and elliptic operators is the following: let $L$ be an elliptic operator of order $l$, then there is some $C > 0$ such that for all $u \in H^{s+l}$ ...
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Two distinct geodesics joining two points on a compact manifold

This is a problem from the book Gallot, Hulin, Lafontaine: Riemannian geometry (3rd edition). Exercise 2.118: For a compact Riemannian manifold, let $p,q$ two points such that $d(p,q) = ...
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The computation of the Laplacian of the heat kernel on a Riemannian manifold

From John Roe, Elliptic Operators, topology and asymptotic methods , page 99 Let $M$ be a manifold of dimension $n$ with fixed point $q$. Let a geodesic local coordinate system $x^{i}$ originate from ...
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Minimizing geodesics don't have kinks

I'm working in a Riemannian manifold where all pairs of points are connected by a minimizing geodesic (i.e. a geodesic whose length equals the distance between the points). Here geodesics are ...
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111 views

Constant curvature geodesic circles on a surface with constant Gauss curvature

Referring to: Curvature of geodesic circles on surface with constant curvature, Is it possible to combine further the last two of the three equations in the link given above into a single ODE / PDE ...
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68 views

Are the topology of a manifold and the topology induced by the metric of a manifold the same?

I am a physics student trying to understand in a rigorous way what a manifold is so please bear with me. Ok, so I am just learning what a topology is and from what I have understood up till now is ...
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52 views

What is the adjoint of the connection operator on a Clifford bundle?

From Elliptic Operators, topology and asymptotic methods, John Roe, page 43-45. Let $M$ be a Riemannian manifold. Let $S$ be a Clifford bundle over $M$, such that each $S_{m}$ over $m\in M$ is a ...
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What meaning did Riemann assign to $dx$?

Detlef Laugwitz wrote a monumental biography of Riemann. The book was translated into English by Shenitzer. Laugwitz discusses Riemann's fundamental essay Uber die Hypothesen, welche der Geometrie ...
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An assumption used to derive the curvature tensor for Riemannian submersions

I was reading the literature about Riemannian submersions, and I came across the result showing the relation between the curvature tensor $\bar{R}$ in a manifold $M$ and the curvature tensor $R$ in a ...
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Is there any difference between a flat manifold and an affine space?

What is the difference, if any, between a flat manifold (in which the Riemann tensor vanished identically) and an affine space?
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Is a Riemannian metric a $2$-form?

In Lee's Riemannian Manifolds; An introduction to Curvature, he defines a Riemannian metric as an element of $\Gamma(T^2_0M)$, a $(2,0)$-tensor. Is this the same thing as a $2$-form? Is there a ...
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manifolds with similar extrinsic and intrinsic distances ( locally)

Is there any specific name for those manifolds caracterized by having (locally) similar ( in some epsilon sense) extrinsic and intrinsic distances?
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Why is the matrix of a Riemannian metric positive definie?

Maybe I could post this as a linear algebra problem but I'll give some context. I know that if $(U, x_1, \ldots, x_n)$ is a local chart of a smooth manifold $M$ I can write locally a Riemannian ...
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Does the conformal class of a complex projective curve contain the Fubini-Study metric?

Let $X \subset \mathbb CP^2$ be a complex curve with metric $g$ induced by the Fubini-Study metric on $\mathbb CP^2$. Since in the case of two-dimensional real manifolds a complex structure is ...
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34 views

Example of complete not-connected riemannian manifold

Are there examples of complete Riemannian manifolds which are not connected ? This question follows my previous question. The more I think about it and the less I'm convinced it exists.
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Riemannian metric and geodesic

For all $p \in \mathbb{R} \smallsetminus \lbrace 0,1 \rbrace$, let $\displaystyle M(p)=\frac{1}{p^{2}(p-1)^{2}}$. Then, let $g_{p} \, : \, (u,v) \, \longmapsto \, uM(p)v$. I am not sure about the ...
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47 views

When do isometries commute with the compatible derivative operator on a semi-Riemannian manifold?

Let $M$ and $\tilde{M}$ be smooth manifolds, each with a metric $g_{ab}$ and $\tilde{g}_{ab}$, assumed here to be smooth symmetric invertible tensor fields, which are non-degenerate but not ...
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81 views

Ricci Tensor, Curvature and Scalar Curvature computation from definition

I am studying a little of Riemannian geometry and I am having some problem in making the connection between two expressions of Ricci tensor, curvature and scalar curvature. Well, in the book that I am ...
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54 views

Question on Torsion free condition for Levi-Civita connection

I was watching a video on Riemannian Geometry. The lecturer mentions that given the defining condition for a connection on a Riemannian manifold $M$ i.e. : $$\nabla_X(Y) : \chi(M) \times \chi(M) \to ...
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Parallel Transport of Geodesic Velocity Vectors

Given a Riemannian manifold $M$ with Riemannian metric $g_{x}:T_{x}M\times T_{x}M\rightarrow\mathbb{R}$ and distance $d:M\times M\rightarrow\mathbb{R}$ determined by length of minimizing geodesics, ...
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251 views

Non-ellipticity of Yang-Mills equations

Let $D=\text{d}+A$ be a metric connection on a vector bundle with curvature $F=F_D$. How does one prove that the Yang-Mills equations $$ \frac{\partial}{\partial x^i}F_{ij}+[A_i,F_{ij}]=0 $$ from ...
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Levi Civita connection along principal curvature directions

Let $(M,g)$ be a surface that can be immersed into $\mathbb{R}^3$. Denote by $\nabla$ the associated Levi Civita connection. Further, let $X_1,X_2$ be the directions of principal curvature which are ...
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Difference betwee parameterization and embedding of manifolds

What is the difference between embedding and parameterization? Why, for example, we say Gauss parameterization of a convex hypersurfaces, and we don't call it an embedding?
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Riemannian manifolds are metrizable?

I've seen lots of casual claims that Riemannian manifolds (even without assuming second-countability) are metrizable. In the path-connected case, we can use arc-length to create a distance function. ...
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Easy solution to Yamabe problem for surfaces

The Yamabe problem asks if, given a Riemannian manifold $(M,g_0)$, it is possible to find a conformal metric $g$ on $M$ with constant scalar curvature. I would like to know if there is some "easy" ...
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extension of a local orthonormal frame on a hypersurface

Let $N$ be a $(n+1)$-dimensional Riemannian manifold and $M\subset N$ a Riemannian hypersurface (embedded or immersed). Let $M$ and $N$ be oriented and choose a unit normal vector field $\nu$ along ...
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relative sign in Hodge star of tensor product

Let $V$ be a vector space of arbitrary (finite) dimension and let $(V, \langle \ ,\ \rangle, I) = (W_1, \langle\ ,\ \rangle_1, I_1) \oplus (W_2, \langle\ ,\ \rangle_2, I_2)$ be a direct sum ...
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What is the meaning of the symbol $\nabla^k$?

In T.Aubin's book, a course in differential geometry, he write the formula $\Delta f=-\nabla^k\nabla_kf$ on a Riemannian manifold, but he never define the symbol $\nabla^k$. It seems that the notation ...
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Riemannian Metric Notation

I am just being introduced to Riemannian metrics, and I am having a bit of confusion on the notation. When reading, I've encountered some different notation in different sources, so I want to make ...
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Gradient of norm of embedding

Let $\varphi:(M,g,\nabla)\to\mathbb{R}^n$ be a smooth embedding of a convex hypersurface. I want to explicitly calculate $$\langle \varphi,\varphi_{\ast}(\nabla\|\varphi\|^2)\rangle.$$ In particular, ...
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How do we check conformal equivalence of parametrized surfaces, e.g. parallel surfaces?

Suppose we have two parametrized surfaces in $\mathbb{R}^3$: $$ X,Y:\mathbb{R}^2 \rightarrow \mathbb{R}^3 $$ The induced metric on either surface is the pullback of the Euclidean metric $\bar g$ due ...
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What distinguishes elliptical coordinates from polar coordinates?

I am trying to identify what characteristic distinguishes elliptical coordinates from polar coordinates. For concreteness, let's write down the expressions. Polar: $$ x=r \cos(t) \\ y=r \sin(t) $$ ...
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44 views

Connected components of the complement of a closed geodesic on a hyperbolic surface.

Let $M$ be homeomorphic to a 2-sphere with a finite number $\geq 3$ of points removed. This implies that $M$ can be equipped with a complete, finite area hyperbolic metric. I imagine $M$ as an ideal ...
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Function whose gradient is of constant norm

Let $f:\mathbb R^n\rightarrow \mathbb R$ be a smooth function such that $\|\nabla f(x)\|=1$ for all $x\in \mathbb R^n$ and $f(0)=0$. I would like to prove that $f$ is linear. I first looked at the ...
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Does the possibility of linear coordinate changes imply that the manifold is Euclidean?

Question. Let $M$ be a smooth manifold admitting an atlas $\mathcal{A}$ (i.e. a collection of coordinate charts that covers the whole manifold) with the following property: for every pair ...
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Interior of a Dirichlet domain in a Riemannian manifold

Let $X\neq\varnothing$ be a complete connected Riemannian manifold. Suppose $G$ is a group of isometries of $X$, acting properly discontinuously on $X$. We assume there is a point $x_0\in X$ such that ...
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What is the exterior normal to the boundary of a Riemannian manifold?

Let $(M,g)$ be a Riemannian manifold with boundary $\partial M$. Let $p \in \partial M$ and in local coordinates $(x_1,\ldots,x_n)$ near $p = (0,\ldots,0)$ the manifold $M$ is given by $\{x_n ...
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Question about a particular estimate in Riemannian geometry.

I have been studying the book Some Nonlinear Problems In Riemannian Geometry - Thierry Aubin. On page $46$ he begins the proof of the Sobolev imbedding theorem to manifolds. The proof is divided in ...
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63 views

Constant in Sobolev-Poincare inequality on compact manifold $M$; how does it depend on $M$?

Let $M$ be a smooth compact Riemannian manifold of dimension $n$. Let $p$ and $q$ be related by $\frac 1p = \frac 1q - \frac 1n$. There is a constant $C$ such that for all $u \in W^{1,q}(M)$ ...
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Reference for construction on Riemannian Manifolds

I would like to know a book or an article where the connected sum of Riemannian manifolds is explained with some detail. I roughly know how to do the construction but I want to be able to check a more ...
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101 views

Riemannian Submerssion

I am reading John Lee's Riemannian Geometry Chapter 3, and I want to do some exercises. I think that I need some hints to solve the following: (Problem 3-8 of that book) Suppose $M$ and $N$ are ...