(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 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 ...
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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 ...
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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 ...
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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 ...
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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 $\mathbb{R}^n$? ...
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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 ...
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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?
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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 ...
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75 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 ...
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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 ...
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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 ...
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Cohomology in Differential Geometry

Below is a communicative diagram: $$\begin{array}[c]{ccc} ...
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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 ...
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50 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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Show that a parallel field has constant length.

Show that a parallel field has constant length (Riemannian-geometry). It is true for all connections?
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34 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 $\mathbb{R}^{n}$
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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 $r \rightarrow 0$, ...
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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 ...
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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 ...
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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 ...
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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 ...
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60 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 ...
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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 ...
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142 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 ...
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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 ...
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“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!
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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, ...
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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 ...
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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 ...
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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 ...
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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 ...
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43 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!
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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 ...
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35 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 ...
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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 ...
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Lifting Riemannian metrics on principal bundles

Given a principal bundle $\pi:M\rightarrow M/G$, there are natural maps $$\pi_{\mathcal{F}}:\mathcal{F}(M)^G\rightarrow\mathcal{F}(M/G)$$ ...
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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*} ...
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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 ...
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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 ...
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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 ...
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Basic misunderstanding of the theorema egregium

The theorema egregium demonstrates that the Gaussian curvature, $K$, is an intrinsic property. What I think this means is that if you know the metric corresponding to the surface, then you can compute ...
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Computing distance on a sphere

Let's say I want to compute the distance between two far points on Earth, say Toronto and Brazil. I can do this by getting in my car, setting my odometer to zero and then driving to Brazil. For me, ...
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for a compact manifold $M$, is the dual space of $H^1(M)$ equal to $H^{-1}(M)$?

Let $M$ be a compact Riemannian manifold. Is it true that $$(H^1(M))^* = H^{-1}(M)?$$ is there some intuitive explanation why? Or some reference? Thanks Here $H^1$ is the usual Sobolev space of $u ...
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Conditions on the metric function in a flat manifold

It is well known that a manifold is flat iff its Riemann tensor vanishes identically. However, the equation $R^{\mu}_{\nu\rho\sigma}=0$ is a differential equation for the metric tensor $g_{\mu \nu}$. ...
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Integration by parts (Differential Geometry)

I am studying the proof of a theorem and I am stuck. It says that by integration by parts we get that: For $g(t)$ a variation of Riemannian metrics wih $g'(0)=h,$ $$\int_{M} (-\Delta (tr h) + ...