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 ...

learn more… | top users | synonyms

0
votes
0answers
34 views

on isometric group

Let $G_n$ be a Lie group, $g$ denotes the left invariant Riemannian metric on $G$. I want to ask for help that how to prove this conclusion: if all principal Ricci curvature of $(G_n, g)$ are ...
2
votes
1answer
70 views

Sufficient condition for $M$ to have constant curvature

I decided to keep my original question. However, I'm having trouble only in a part of it (check NOTE) Let's consider a Riemannian manifold $(M,g)$, with the Levi-Civita connection $\nabla$. I would ...
1
vote
1answer
40 views

Boundary points of a manifold

I'm reading about Riemannian Geometry and my question is regarding Manifolds with Boundary. I want to show a point of a manifold with boundary is either an interior point or a boundary point, so no ...
1
vote
0answers
52 views

Order of Riemann tensor indexes and the Ricci Identity

I have seen the Ricci identity written variously as $R_{ijk}{}^l x^k = (\nabla_i\nabla_j- \nabla_j\nabla_i) x^l$ $R_{ij}{}^l{}_k x^k = (\nabla_i\nabla_j- \nabla_j\nabla_i) x^l$ $R^l{}_{kij} x^k = ...
2
votes
0answers
40 views

When does a pseudo-Riemannian manifold have an always positive norm Killing field?

When does a pseudo-Riemannian manifold have an always positive norm Killing field? (you may assume that the isometry group is of the form $SO(1,n)$ if necessary) In the context of general ...
0
votes
0answers
66 views

The level set of Lipschitz functions

Suppose $u$: $R^N\to R$ is lipschitz, then do we have a.e. level set of $u$ has Lipschitz boundary? Is this anything to do with Sard theorem? Sard theorem states that a.e. Level set of smooth ...
2
votes
0answers
67 views

At what conditions a compact metric space of covering dimension $n$ (on $\mathbb R^n$) is an n-manifold?

In the discussion to the MSE post an answer of @MatthewPancia with correction in a comment of @JasonDeVito would state: Every compact metric space of covering dimension $n$ can be embedded ...
2
votes
2answers
63 views

Raising and Lowering Through Differentiation

I'm calculating the Christoffel symbols of the second kind which is of course defined as multiplying the symbol of the first kind multiplied by the contravariant metric. I was thinking of how to make ...
0
votes
1answer
19 views

A question about sum of angles in a non-positive curvature Riemannian manifold

Suppose on a non-positive curvature Riemannian manifold,we have a geodesic triangle $\triangle abc$ ,and counterpart edges donates $\alpha,\beta,\gamma$. If now I get $$ a^2 \geqq b^2+c^2-2bc ...
2
votes
1answer
32 views

Ricci curvature version of Cartan-Hadamard theorem?

Is the following assertion true : If $M$ is a simply-connected manifold with $\operatorname{Ric}<0$ (or $\operatorname{Ric}\leq -k$ for $k$ positive) then $M$ is diffeomorphic to $\mathbb{R}^n$? ...
1
vote
2answers
68 views

Is Relativity a specific instance of Riemannian geometry?

If I am a mathematician and do not anything about Special/General Relativity, then should I study Riemannian geometry to learn Relativity? Is Relativity just an instance/example of some particular ...
2
votes
1answer
39 views

Explicit example of a compact manifold of dimension $>2$ with strictly negative sectional curvature

I am looking for examples of compact manifold of dimension $>2$ with strictly negative sectional curvature (for dimension 2 it is well-known). Can anybody please help?
3
votes
1answer
54 views

Symmetry of Killing Vectors in Covariant Derivative

Several times, I've seen statements along the lines of "$\nabla_X Y=\nabla_Y X$ because $X$ is a Killing vector field." One example I found on Stack Exchange is here. I have yet to understand why ...
1
vote
1answer
88 views

Smoothly homotoping a sphere in $\mathbb{R}^3$

Start with the standard sphere $S^2$ and consider another (diffeomorphic) sphere $S$ such that there is a family of deformations of $S^2$ in $\mathbb{R}^3$ that ends in $S$. If $S$ is positively ...
2
votes
1answer
47 views

Conformal transformation of the divergence

Let $f$ be a smooth function on a $n$-dimensional Riemannian mainfold $(M, g)$, so that $\tilde{g} = e^{2f} g$ is a conformal transformation of $g$. I'm trying to show that the divergence operator ...
3
votes
1answer
89 views

Riemannian curvature tensor of product manifolds

Let $(M_{1},g_{1})$ and $(M_{2},g_{2})$ be two Riemannian manifolds. Let $% R_{1}$ and $R_{2}$ be the (1,3)-type Riemannian curvature tensors of $M_{1}$ and $M_{2}$, respectively. Finally, let $R$ be ...
2
votes
0answers
78 views

Cohomology in Differential Geometry

Below is a communicative diagram: $$\begin{array}[c]{ccc} ...
4
votes
0answers
42 views

What properties do isospectral Riemannian manifolds share?

I'm studying the Laplacian on (compact) Riemannian manifolds, and it turns out that if the Laplacian operators of two such spaces share their spectrum (the spaces are then called isospectral), then ...
3
votes
1answer
58 views

Jacobi field and the metric

I'm reading about Jacobi fields lately, and have noticed some features of it (and it's derivative) with respect to the metric. Thinking about that, I had an non-based, purely intuitive thought that ...
3
votes
0answers
60 views

Closed geodesic loop on compact manifold

Let $M$ be a compact manifold (hence complete). Let $p$ be any point on $M$. Is it true that we can always find a geodesic loop based at $p$? If $M$ is non-simply connected it is true as each ...
0
votes
0answers
24 views

Explicit formula for the (n-2)th derivative of the Jacobi equation

The $n-2$ order derivative of the Jacobi equation is given by: $$\frac{D^n}{dt^n} V_i+\sum\limits_{l=0}^{n-2} \binom{n}{k} (\nabla_{\gamma '}^{(n-2-l)}R)(\gamma ' ,\nabla_{\gamma '}^{(l)} V_i)\gamma ...
3
votes
0answers
51 views

Surfaces (with boundary) in $\mathbb{R}^3$ conformal to the cylinder

Consider the usual cylinder $S^1 \times [0, 1]$ embedded in $\mathbb{R}^3$. I am interested in knowing what are the surfaces in $\mathbb{R}^3$ that are conformal to this cylinder. If this were a ...
2
votes
1answer
86 views

How to show that geodesics exist for all of time in a compact manifold?

Let $M$ be a compact manifold and the tangent bundle $TM$ have a Riemannian metric $g$ so that it is isomorphic to the cotangent bundle $T^*M$. Consider the pull-back of the canonical symplectic form ...
6
votes
1answer
95 views

Covariant derivative of vector field along itself: $\nabla_X X$

Consider a vector field $X$ on a smooth pseudo-Riemannian manifold $M$. Let $\nabla$ denote the Levi-Civita connection of $M$. Under which conditions can something interesting be said about the ...
0
votes
1answer
27 views

Show that a parallel field has constant length.

Show that a parallel field has constant length (Riemannian-geometry). It is true for all connections?
1
vote
1answer
37 views

Find an example of n-dimensional differentiable manifold

Find an example of $n$-dimensional differentiable manifold whose points are not points of the variety $\mathbb{R}^{n}$
0
votes
0answers
24 views

Metric in normal coordinate

Using Gauss’s lemma we can write the metric in normal co-ordinate as $g(r, θ) = dr^2 + r^2h_{ij}(r, θ)dθ^i ⊗ dθ^j$ (where metric on $S^{n-1}$ is $\tilde {g}=dθ^i ⊗ dθ^i$). Now as $r \rightarrow 0$, ...
1
vote
0answers
97 views

Showing a property of a curvature tensor in $S^2$

Consider $S^2 \subset \mathbb{R}^3$. I need to show that if $$R_{ijkl} = -g(R(\partial_i,\partial_j)\partial_k,\partial_l)$$ is a curvature tensor in $S^2$ and $g$ is a metric also in $S^2$, then ...
1
vote
0answers
38 views

Variation of geodesic and Jacobi field

I was reading on Jacobi field of a geodesic, and noticed that given a geodesic $\gamma$ it is defined using the term of variation or family of geodesics $\gamma_s$ but never mentioned how to create ...
0
votes
1answer
50 views

Are there “interesting” examples of complete finite-volume non-compact Riemannian manifolds that are not non-positively curved?

Moreover, it would be good that, if $M$ is such an example, its universal cover $\tilde M$ is "highly" symmetric, i.e. the group G of isometries of $\tilde M$ is "big" (for example, transitive, or a ...
0
votes
1answer
51 views

Geodesic loops in Riemann homogeneous spaces

Let M be a Riemannian homogeneous space, i.e. the isometry group acts transitively. Prove: any geodesic loop (with possible angle at the starting point) is a closed geodesic (smooth at the starting ...
2
votes
1answer
125 views

Sectional curvature of product metric?

If $M$ and $N$ are Riemannian manifolds, can we relate the sectional curvature of the product Riemannian manifold $M \times N$ to those of $M$ and $N$? If both $M$ and $N$ have non-negative (or ...
0
votes
2answers
66 views

Riemann manifolds in relation to other classes of differentiable manifolds

I am trying to get an overview over the different categories of manifolds. In particular i have the following chain of inclusions: Riemann surfaces $\subset$ complex manifolds $\subset$ orientable ...
6
votes
1answer
176 views

Curvature of De Sitter's space: where does the sign comes?

Consider $\Bbb L^3 = (\Bbb R^3, {\rm d}s^2)$, where: $${\rm d}s^2 = {\rm d}x^2 + {\rm d}y^2 - {\rm d}z^2.$$ We have both the hyperbolic space: $$\Bbb H^2(-1) = \{(x,y,z) \in \Bbb L^3 \mid ...
1
vote
1answer
15 views

Out of plane cross section evolution of surfaces based on local geometry information

With this question I would like to kindly ask for feedback or general pointers to even remotely related works in regards to a challenge I face. Given a smooth surface $S$ $:\mathbb{R}^2\rightarrow ...
0
votes
1answer
64 views

“Rigid” Riemannian metrics

What do we mean when we say that a Riemannian metric $g$ is rigid? For example, the Eguchi-Hanson metric is rigid as an Einstein metric. Any help is appreciated!
3
votes
2answers
61 views

Resource for learning about the Laplacian on Riemannian manifolds

Does anyone have any recommendations for, as the title suggests, a book from which to learn about the Laplacian on Riemanian manifolds, or even just on smooth manifolds? I found this presentation, ...
3
votes
1answer
44 views

Symmetries of Weyl Tensor

We know that the Riemannian curvature $(0,4)-$tensor may be decomposed as $$Rm=W\;+\;A *g$$ where $*$ is the Kulkarni-Nomizu product, and $A$ the Schouten tensor. I am studying the proof of a theorem ...
4
votes
1answer
67 views

Green's operator for elliptic differential operator

Let $$ P:\Gamma(E)\rightarrow\Gamma(F) $$ be an elliptic partial differential operator, with index zero and closed image of codimension 1, between spaces $\Gamma(E)$ and $\Gamma(F)$ of smooth sections ...
0
votes
1answer
29 views

Volumes and metric

I have a pretty general question regarding volumes of manifolds and metrics. I was wondering if knowledge of the different volumes and how they relate to each other can tell you anything about the ...
2
votes
1answer
38 views

What is a minimal fiber of a Riemannian submersion

I am reading "Spectral Geometry, Riemannian Submersions and the Gromov-Lawson conjecture" by Gilkey, Leahy and Park, and I'm having some trouble with some of the terms they introduce without ...
1
vote
1answer
48 views

A riemannian metric is in a neighborhood of $g$ (?)

What do we mean by this: "A Riemannian metric $g_1$ is in a small $C^{l+1,\alpha}$ neighborhood of the metric $g$" ? Any help is appreciated!
1
vote
0answers
18 views

Can some components of metric be Fisnlerian while the others be Riemannian?

A Finsler metric reduces to a Riemann metric in case it loses its dependence on velocities. Now, my question is this: Can we have a Finsler metric in which some components of the metric have velocity ...
1
vote
1answer
44 views

Isometries of a Connected Surface

Let $S \subset \mathbb{R}^3$ a connected surface, $p \in S$ and let $f,g:S \rightarrow S$ be two isometries. Suppose that $f(p)=g(p)$ and $d_p f (X)= d_p g(X)$, for all $X \in T_pS$. I want to proof ...
3
votes
2answers
132 views

Good references on Riemannian Geometry

I'd like a textbook that covers do Carmo's contents (can be more), but that isn't do Carmo. I did not like his writting style. That being said, I particularly like the styles of: Walter Rudin ...
4
votes
1answer
50 views

Lifting Riemannian metrics on principal bundles

Given a principal bundle $\pi:M\rightarrow M/G$, there are natural maps $$\pi_{\mathcal{F}}:\mathcal{F}(M)^G\rightarrow\mathcal{F}(M/G)$$ ...
1
vote
3answers
85 views

Prove: Gravitation operator is invertible.

Let $(M,g)$ be a Riemannian manifold and $\Gamma(S^2M)$ the space of symmetric 2-covariant tensors. Define the gravitation operator as the map \begin{align*} ...
3
votes
0answers
31 views

complete vector field on Riemannian manifold with lower bound

From do Carmo's Riemannian Geometry P151: Let M be a complete Riemannian manifold, and let $X$ be a differentiable vector field on $M$. Suppose that there exists a constant $с > 0$ such that ...
2
votes
1answer
49 views

Is the on-diagonal heat kernel “local” with respect to the metric?

Question Let $X$ be a manifold, and $\mu_A$, $\mu_B$ two Riemannian metric on it which agree on an open subset $U\subset X$, i.e. $\mu_{A\,|U} = \mu_{B\,|U}$. Let $K_A(t;z,w)$ resp. $K_B(t;z,w)$ be ...
1
vote
0answers
51 views

Heat Equation on Manifold

Laplacian operator is defined well on Riemannian manifold, denoted by $\Delta$. Therefore people can study PDE $\Delta f=0$ on manifold. So is there any analogy to heat equation or wave equation on ...