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# Questions tagged [riemannian-geometry]

For questions about Riemann geometry, which is a branch of differential geometry dealing with Riemannian manifolds.

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### Can the Shape Operator be defined on arbitrary Riemannian manifolds?

While reading Chapter 15 of Needham's Visual Differential Geometry (picture of the passage at the end of the post) I came to wonder a couple of things: What is the rigorous definition of the Shape ...
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### On writing integration on Riemannian submanifolds in terms of exponential map

I have a basic question about integrating (restriction of) functions to a Riemannian submanifold. Particularly, given a Riemannian manifold $M$, an embedded Riemannian submanifold $N \subseteq M$, and ...
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### Integral over Curvature and metric components

Let $(M,g)$ a closed connected oriented manifold of dimension $n=2m$. Furthermore, let $\nabla$ be the Levi-Civita connection and $R$ is the Riemannian curvature tensor. Let $h$ be a symmetric $(0,2)$-...
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### Fields related to differential geometry

I've been reading introductory texts in differential geometry, and I'm wondering what comes next. The most natural next step seems to be Riemannian geometry, but I also know that differential ...
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### In a geodesic triangle, is the longest side opposite to the largest angle?

If I have a complete (smooth) Riemannian manifold $(M,g)$ and three points on it, that I connect with distance minimizing geodesics, will the longest edge be opposite to the largest angle? In ...
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### Gauss map and its differential

Good morning, I am looking for help to understand a concept related to the Gauss map $$N: M \rightarrow S^2$$ Since I am interested in the variation of $N$, I evaluate the differential and do not ...
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### Positive scalar curvature Einstein manifolds are noncollapsed

I am currently working through some of the exercises in Ricci Solitons in Low Dimensions by Bennett Chow, and I've been stuck on Exercise 1.23. It asks you to prove that positive scalar curvature ...
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### Existence of smooth triangulation for Riemannian 2-manifold

Most proofs that I can find of the Gauss-Bonnet Theorem for a compact Riemannian $2$-manifold $M$ always start with the assumption that $M$ has a smooth triangulation, i.e. a triangulation where the ...
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### Is squared geodesic distance about a point convex when defined on a convex ball centered at that point?

Let $\mathcal{M}$ be a complete Riemannian manifold. Let $x \in \mathcal{M}$, and let $r_x >0$ be the convexity radius at $x$. Let $B \subset \mathcal{M}$ be a geodesic ball centered at $x$ with ...
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### The image of a Riemannian manifold under a distance-based mapping to $\mathbb{R}^n_{\geq 0}$

Consider $n$ points, $p_i\in M,\, i=1,\ldots , n$, in a Riemannian manifold $M$, and define the map \begin{array} &f: M&\to& \mathbb{R}^n_{\geq 0}\\ \qquad q &\mapsto& \left(\begin{...
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### Dual of the sum of dual metrics

Suppose I have a (smooth, connected, compact) Riemannian manifold $M$ and two smooth metric tensors $g_1$ and $g_2$. I can define the metric tensor $g:= (g_1^\star+g_2^\star)^\star$ where for a tensor ...
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### Vector field of rotation around an axis in spherical coordinates

This is a qualifying exam question in Differential Geometry. I'm new to the subject and am reading up from John Lee's Introduction to Smooth Manifolds. Let $M=S^2\subset\mathbb{R}^3$ be the unit ...
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### Existence of some point $t' \in [0,b] \xrightarrow{\gamma} U_0 \subsetneq U$ ( unit speed radial geodesic ) satisfying certain property.

Let $(M,g)$ be a Riemannian $n$-manifold. $p\in M$. Let $U$ be a normal neighborhood of $p$, and $r$ be the radial distance function on $U$ ( C.f. Def. : John Lee's Introduction to Riemannian ...
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### Fundamental equations of a Riemannian immersion

The Gauss, Codazzi-Mainardi and Ricci equations are the three fundamental equations of a Riemannian immersion. The Gauss equation inputs 4 tangent vectors, the Codazzi-Mainardi inputs 3 tangent ...
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### Three simple(?) questions about proof of Lemma 10.18 in the John Lee's Riemannian manifold book.

I am reading the John Lee's Introduction to Riemannian manifold, Proof of the Lemma 10.18 and stuck at some statements : Lemma 10.18. Suppose $(M,g)$ is a Riemannian manifold with parallel curvature ...
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### A curve in a single orbit in the space of Riemannian metrics

Let $M$ be a smooth manifold. Let $\mathcal M$ denote the collection of all Riemannian metrics on $M$. There is a right action of the product group $\mathbb R^+ \times \text{Diff}(M)$ on $\mathcal M$ ...
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### For every n-Riemannian manifold, is every local chart of nonzero radius isometrically embeddable in (n+1)-Euclidean space as Monge patch?

We hear common phrases such as "3-dimensional space curves into 4th dimensional flat space" being thrown around popular science media. And I want to confirm whether such statement is true. ...
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### Relation between Christoffel Symbols and metric tensor

I am learning about Riemannian geometry, and I have a question about relation between the Christoffel symbols and the metric tensor of the manifold. Is it true that the Christoffel symbols are unique ...
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