(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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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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1answer
22 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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34 views

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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14 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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3answers
133 views

How to define intrinsic curvature?

I have been exploring differential geometry slightly... And i'm trying to grasp how to define intrinsic curvature, from a visual/geometric viewpoint... One formulation I got was that Intrinsic ...
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1answer
70 views

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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1answer
32 views

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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1answer
31 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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41 views

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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2answers
66 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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1answer
22 views

Local geodesics in uniquely geodesic spaces

Suppose $Y$ is a proper, uniquely geodesic metric space. In such a space, is any local geodesic in fact a geodesic? Here the terms "geodesic" and "local geodesic" are taken in the metric sense: a ...
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1answer
40 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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2answers
137 views

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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1answer
59 views

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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32 views

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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1answer
64 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 ...
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2answers
241 views

Hilbert theorem and constant negative curvature surfaces

Let us consider the tractroid (pseudosphere) obtained by rotation from the tractrix curve. The surface is not defined on the "big rim", so it is not a complete set. Hilbert's theorem states that there ...
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227 views

Hodge-Star-Operator on arbitrary oriented basis

Assume that $V$ is oriented finite dimensional vectorspace with dimension $n$, $g \in T^0_2(V)$ a given symmetric and nondegenerate tensor. Let $\mu$ be the corresponding volume element of $V$. ...
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36 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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1answer
40 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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0answers
40 views

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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2answers
242 views

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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1answer
52 views

Minimum ratio between surface and volume in a riemannian manifold

In an euclidean three - dimensional space the sphere is the geometric figure with the minimum ratio $R=\frac{S}{V}$ with $S=4\pi r^2$ and $V=\frac{4}{3}\pi r^3$, so we have: $$R=\frac{1}{3}r$$ where ...
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3answers
85 views

Riemannian metric, compute

I have a question that may look for you as silly. A few years ago I took a course of Riemannian geometry. Well, the first problem I found is to understand the generalization of tangent plane (in ...
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1answer
48 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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1answer
24 views

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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2answers
66 views

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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1answer
125 views

Product neighborhood theorem with boundary

The Product Neighborhood Theorem states that if $N\subseteq M$ is a smooth compact submanifold without boundary of codimension $n$ and there is a trivialization of the normal bundle (wrt. some smooth ...
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1answer
59 views

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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2answers
238 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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0answers
14 views

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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1answer
37 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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39 views

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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0answers
18 views

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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1answer
28 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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2answers
60 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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2answers
149 views

Definition of the principal symbol of a differential operator on a real vector bundle.

I'm trying to understand the construction of the dirac operator on a manifold, but actually I guess that doesn't really matter for the question at stake. I'm interested in understanding a definition ...
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1answer
31 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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1answer
41 views

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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1answer
75 views

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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44 views

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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1answer
31 views

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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2answers
78 views

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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1answer
45 views

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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1answer
69 views

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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1answer
41 views

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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1answer
38 views

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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1answer
68 views

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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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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51 views

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 ...