# Tagged Questions

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 ...

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### Why are geodesically convex sets diffeomorphic to $\Bbb R^n$?

In the construction of a good cover for a manifold $M^n$, Bott & Tu use the fact that each point in $M$ is contained in a geodesically convex set (after picking a Riemannian metric). They then ...
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### sectional curvature of hyperbolic space

For a pair of $(X,Y)$ of linearly independent vectors in $T_pM$, $p\in M$, the sectional curvature is defined as $$K_p(X,Y)= - \frac{<R(X,Y)X,Y>}{|X|^2 |Y|^2 - <X,Y>^2}$$ The problem I'm ...
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### Uniqueness of minimizing geodesic $\Rightarrow$ uniqueness of connecting geodesic?

Let $M$ be a complete connected Riemannian manifold. Fix $p \in M$. Assume every point in $M$ has a unique minimizing geodesic connecting it to $p$. Is it true that for every point, the only ...
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### A exercise of Riemannian geometry . [on hold]

In picture below,I don't know how to start the second question . It is obvious that the isometry of $R^3$ keep the dimension , so there exist such isometry. But seemly, it is too simple . Besides, ...
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### How to prove that two compact 1-dim Riemannian manifolds with same length must be isometric?

How to prove that two compact 1-dim Riemannian manifolds with same length must be isometric ? I know the compact 1-dim manifold must be homeomorphism to $S^1$ , but how to do a specific isometric ?
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### About geodesics in product manifolds

My question is about the answer to this post : Geodesics on the product of manifolds If $(M_{1},g_{1})$, $(M_{2},g_{2})$ are Riemannian manifolds and if $\nabla^{1}$ (resp. $\nabla^{2}$) denotes the ...
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### sectional curvature, ricci tensor and scalar curvature of the hyperbolic space [closed]

Who can help me to compute the sectional curvature, Ricci tensor and the scalar curvature of the hyperbolic space $H^3$ ? Thanks!!
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### A Riemannian manifold with constant sectional curvature is Einstein. [closed]

A Riemannian manifold with constant sectional curvature is Einstein. Why? It's true the inverse?
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### Why a Riemannian manifold minus one point is not complete? [closed]

Could you give me a proof that a Riemannian manifold minus one point is ever complete? Thanks!!
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### Covariant Taylor series

I am reading the following lecture notes of Avramidi https://www.researchgate.net/publication/255565392_Analytic_and_geometric_methods_for_heat_kernel_applications_in_finance I want to understand ...
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### Misunderstanding of Atiyah-Singer

I just looked up the Atiyah-Singer theorem and by ignoring technical details I had the impression that it tells us that any elliptic operator on a compact manifold satisfies Analytical index = ...
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### Covariant derivative of parallel transport

I am learning Riemannian geometry and don't get why the following is true. We are on a Riemannian manifold with the Levi Cevita connection $\nabla$. Let $\mathcal{P}(x,x')$ be the parallel transport ...
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Let $S_1, S_2 \subseteq \mathbb{R}^n$ be two linear $k$-dimensional subspaces. Does there always exist a hyperplane $H$ such that $S_1 = R_H S_2$, where $R_H$ denotes the orthogonal reflection across $... 1answer 27 views ### Integration of differential form on ellipsoidal surface with singularity in origin As picture below ,I want to compute the (2) , because there is a singularity in$\{0\}$and$\omega$is closed . So ,I have $$\int_M\omega=\int _{\partial B_1(0)} \omega$$ I think there is a ... 1answer 149 views ### Green's Function for Laplacian on$S^1 \times S^2$As indicated by the title, I am looking to find the Green's function for the Laplacian on$S^1 \times S^2$. Is such a function known? If not, does anyone have an approach to constructing such a ... 0answers 19 views ### integral constraint induce a manifold on Sobolev space given the set $$M:=\{u\in H^2(\Omega):\int_{\Omega}u=m\,\}$$$m\in \mathbb{R}$,$\Omega $is a bounded piecewise smooth domain in$\mathbb{R}^n$. also denote by$u(t)$a map:$u(t):(0,T)\to M$... 0answers 26 views ### Help with derive geodesic equation Let$(M,g)$be a pseudo-riemannian manifold and$p,q\in M$. Suppose$\alpha:[a,b]\to M$a smooth curve on$M$such that$\alpha(a)=p$and$\alpha(b)=q$. If we consider: $$L[H(s,\cdot)]=\int_a^b \sqrt{... 0answers 78 views ### Relearning differential geometry I will shortly describe my situation and than formulate the problem. From around year I am working under supervision of my professor on master thesis in differential geometry (mainly discussion of ... 0answers 61 views ### A parallel transport around the Earth Assume the Earth to be a 2-sphere. I start walking from the point (\theta_0,\varphi_0) around the earth while my body heading north all along the way. In simple words, I walk east/west (It doesn't ... 0answers 16 views ### Is the norm of tensor fields just Hilbert-Schmidt norm/ generalized L^p-norm? As I am rather comfortable with functional analysis language and was new to Riemannian geometry, I am curious when inspecting the norm of Ricci tensor, which is:$$|\mathrm{Ric}|^2=g^{ij}g^{kl}R_{ik}... 0answers 98 views ### How interpret the dual lattice$\Gamma^*$? In the book Eigenvalues in Riemannian Geometry of Isaac Chavel page$28$-$29$, they talk about the lattice$\Gamma$and it is defined as $$\Gamma = \left\{\sum_{j=1}^n \alpha^j v_j : \alpha^j \in \... 1answer 46 views ### Intuition behind definition of K-K asymptotic flatness I am reading some notes on black holes, and am confused by this definition of Kaluza-Klein asymptotic flatness: If a spacetime (M, \mathbf{g}) contains a spacelike hypersurface \mathscr{I}_{ext}... 1answer 81 views ### How is defined the inner product g_p on T_p \mathbb{R}^n/\Gamma at the point p? In the book Eigenvalues in Riemannian Geometry of Isaac Chavel page 28, I have some questions related to the resolution of the spectrum of the tori. The lattice acts on \mathbb R^n by$$γ(x)... 0answers 15 views ### Cauchy horizon of a future Cauchy hypersurface I'm studing on the book Semi-Riemannian geometry by O'Neil. I'm tryng to understand the proof of the Hawking's singularity theorem (theorem 55A in the book). What I don't understand is why if$S$... 1answer 25 views ### Divergent Curves and Complete Manifolds I'm working on a problem in do Carmo's Riemannian geometry book (chapter 7, problem 5). He states that a divergent curve on a noncompact Riemannian manifold$M$is a curve$\alpha: [0, \infty) \to M$... 2answers 36 views ### When are embeddings into Euclidean space unique up to ambient isometry? Suppose I have a Riemannian smooth manifold$M$and a smooth isometric embedding$M \hookrightarrow \mathbb{R}^n$. Is this embedding necessarily unique up to some isometry of$\mathbb{R}^n$? If not, ... 1answer 26 views ### Is the subgroup of homotopically trivial isometries a closed subgroup of the isometry group? Let$(M,g)$be a connected Riemannian manifold. Then according to the Steenrod-Myers-Theorem, the isometry group$\text{Isom}(M,g)$of$(M,g)$is a compact lie group with the compact-open topology. ... 0answers 8 views ### Is the variational field orthogonal to the velocity of the geodesic? Let$\gamma(t)$be a geodesic on a Riemannian manifold. Let$f(s,t)$be a variation about$\gamma$. Is it always true that the variational field$\frac {\partial f}{ \partial s}(0,t)$is always ... 0answers 40 views ### Existence of the lift of a curve Let$(M, g)$be a complete Riemannian manifold and let$(\tilde{M}, \tilde{g})$its universal cover. Let$\pi : \tilde{M} \to M$be the covering map. Let$\gamma : I \to (M, g)$be a smooth curve ... 1answer 27 views ### Line in product mainifold Let$(M_1, g_1)$and$(M_2, g_2)$be two complete Riemannian manifolds and consider the product$(M, g) = (M_1 \times M_2, g_1 + g_2)$. Let$\gamma : \mathbb{R} \to (M,g )$be a line. I can write$t ...
Suppose $M$ is an embedded sub-manifold of $D$-dimensional Euclidean space $E^D$, with embedding $\phi:M \hookrightarrow E^D$. And suppose I know the induced Riemmanian-metric $g$ on $M$, which ...
Let $M$ be a Riemmanian manifold and $X$ be a vector field thereon. My question is why are these two problems equivalent?: \operatorname{argmin}_{\phi}\int_M |\nabla \phi - X|^2 \...