# 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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### Existence of closed, non self-intersecting geodesics on compact manifolds

Let $(M,g)$ be a compact Riemannian manifold. It is well-known that there always exists a nontrivial closed geodesic in $M$ (the so-called Lusternik-Schnirelmann theorem). But such a geodesic could ...
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### $C^k$ one-parameter family of metrics

Consider a smooth Riemannian manifold $M$ and a $C^k$ one-parameter family of Riemannian metrics $g_t$ on $M$. Here $k$ could be any integer, $k$ could be infinity, when the one-parameter family $g_t$ ...
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### converse of theorema egregium

Suppose $(M,g)$ is a $3$ dimensional Riemannian manifold and $N$ is any surface imbedded in $M$. If the theorema egregium holds for $N$ does it follow that $M$ is flat? The way I'm thinking of the ...
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### Long time existence of Ricci flow on compact surfaces of negative curvature

Is there a long time existence for the Ricci flow on compact negatively curved surfaces? I just read that the normalized Ricci flow has a long time solution converging to a metric of constant negative ...
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### Jacobian of a map between two spaces of different dimension.

I have been asking quite a few questions lately on this same topic, but I am really trying to get a grasp of a lot of the content of my Riemannian Geometry course, so I appreciate the help! on p. 14 ...
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### Does bounded scalar curvature imply bounded Ricci curvature?

Does bounded scalar curvature imply bounded Ricci curvature? It is trivial to show the converse, but I do not know whether the above is true. Inspired by a vaguely similar question, I am thinking ...
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### Ricci flow on a compact surface of constant negative curvature

Let us consider a compact surface of constant negative curvature $-1$ and apply the Ricci flow on it. Will the resulting surfaces for short time also have constant negative curvature? If yes, will the ...
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### Does every manifold M always admit a Riemannian metric?

In the book "Geometry and Topology for Physicists" by Nash and Sen, in Section 7.6, after showing that the structure group $GL(n,\mathbb{R})$ of a frame bundle $F(M)$ (for a general manifold $M$ of ...
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### Riemannian generalization/adaption of the Hubbard–Stratonovich transformation

I'd like to write the Hubbard–Stratonovich (HS) transformation of a scalar function on a Riemannian manifold (curved space). This transformation is quite simple in Euclidean space. One can consider ...
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### Intrinsic Riemannian distance vs. Euclidean distance

The intrinsic metric $d_i(x,y)$ on a Riemannian manifold is the infimum over all curves joining $x$ and $y$, of the arc length of these curves. For a surface $S$ embedded in $\mathbb{R}^3$, ...
I need some help with the following. We are in the Heisenberg group $\mathbb H^n$: we denote points $P\in \mathbb H^n$ as $P=(x_1,\dots,x_n,y_1,\dots,y_n,t)$ and let $X_j=\frac{\partial}{\partial x_j}+... 1answer 80 views ### Advanced introduction to riemmanian geometry After some time of studying differential geometry/topology while avoiding riemmanian geometry I find mysef at a wierd position. I'd like to read a book that contains an introduction to riemannian ... 0answers 24 views ### Metric on Heisenberg group arising from CR isomorphism to punctured odd-dimensional sphere It is well known that odd-dimensional spheres$S^{2n+1} \subset \mathbb{C}^{n+1}$are CR manifolds, and on removing a point we get a CR isomorphism to the (underlying manifold of the) Heisenberg group ... 0answers 27 views ### Clarification about the definiton of a metric on a vector bundle Let$E$be a vector bundle over a manifold$M$. By definition, a metric on$E$is a function$g:E \times _M E \to M \times \mathbb{R}$. In wikipedia, they say$g$is a bundle map. However, it is not ... 2answers 112 views ### The arc length in Riemannian geometry is well defined (independent from the choice of the coordinates) Let be$(M,g)$a connected Riemannian manifold and$p,q \in M$. If$ \phi : [a,b] \rightarrow M$is$C^\infty$we define the arc-length of the curve$\phi$as the quantity: $$J(\phi )= \int_a^b f(\... 0answers 39 views ### Isomorphisms of bundles Studying on vector bundles came across the following problem. Question: Let M_1 and M_2 Riemannian manifolds with TM_1^{\perp} and TM_2^{\perp} their normal of ranks k_{1} and k_{2}, ... 1answer 69 views ### Metric compatibility of dual connection Let (M,g) be a Riemannian manifold with Levi Civita connection \nabla. Then \nabla satisfies a compatibility condition: (\nabla_ZX,Y)+(X,\nabla_ZY)=Z((X,Y)) where (\cdot,-) is a Hermitian ... 1answer 43 views ### Laplace-Beltrami operator as sum of orthogonal projections Let M be a submanifold of \mathbb R^l with the induced metric. Let (\xi_\alpha) be the standard orthonormal basis on \mathbb R^l. For each x \in M, let P_\alpha(x) the projection of \xi_\... 1answer 34 views ### Verification of some conditions and facts of the Laplacian on a Riemannian manifold I came across the following introduction to a paper I was reading: "Let L be a second-order, elliptic, differential operator, with smooth coefficients, on a Riemannian manifold M. Assume -L is ... 3answers 186 views ### uses of Riemannian geometry for questions not related to Riemannian geometry Poincaré conjecture (whose formulation does not make use of notions from Riemannian geometry: "Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere") was eventually proved using ... 0answers 44 views ### Uniqueness of covariant derivative in Do Carmo 2.2 Proposition: Let M be a differentiable manifold with an affine connection \nabla. There exists a unique correspondence which associates to a vector field V along the differentiable curve c:... 1answer 93 views ### Metric tensor for n-sphere in ambient coordinates Let S^n be the unit n-sphere embedded in \mathbb{R}^{n+1}:$$ S^n = \{ a \in \mathbb{R}^{n+1} \mid a \cdot a = 1 \} $$What is the induced metric tensor for the sphere, in \mathbb{R}^{n+1} ... 1answer 53 views ### Concavity of distance function in \mathbb{R}^n or determinant of (x^T \cdot x) I would like to compute the concavity of the distance function in \mathbb{R}^n. Let f(x) =- \Vert x \Vert in \mathbb{R}^n. Then \nabla_xf=- \frac{x}{\Vert x \Vert}. And$$-\operatorname{... 1answer 75 views ### Is there a natural Riemannian structure on the total space of a vector bundle? Suppose$B$is a Riemannian manifold and$\pi: E \to B$is a smooth vector bundle equipped with a metric. Is there a natural Riemannian metric on$E$, i.e. a bundle metric on$TE\to E$? It seems ... 1answer 35 views ### Computation of Liebracket for Vectorfields assosiated with a Variation of Geodesics Let$(M,g)$be a Riemannian manifold,$V \subset \mathbb{R}^2$be an open subset and$\alpha: V \rightarrow M; (s,t) \mapsto \alpha(s,t)$a smooth map. for$(s,t) \in V$one can define $$\frac{\... 1answer 111 views ### Volume density on a Riemannian manifold as a measure I am having some trouble in seeing exactly how the Riemannian density form gives rise to a measure on \text{Borel(M)}. Let (M,g) be a Riemannian manifold. We have the Riemannian density \mu_g. ... 1answer 34 views ### points which are fixed points of a finite group action consider an open set \tilde{U}\subset\mathbb{R}^n and a finite Lie-group G, which acts smoothly on \tilde{U}, i.e. we have a smooth map G\times \tilde{U}\rightarrow\tilde{U}. Suppose further, ... 0answers 38 views ### Diffeomorphism and Orientable double cover Suppose that the orientable double cover of two homeomorphic surfaces are diffeomorphic, is it true that these surfaces are diffeomorphic? 0answers 37 views ### One-sided surfaces and the second variation area formula. I know how to find the second variation area formula for a two-sided minimal embedded surface in a 3-manifold and the condition for such a surface to be stable. But, what about one-sided surfaces? ... 3answers 138 views ### Motivations for Hyperbolic Geometry Why would one study hyperbolic geometry? I am only aware of the motivation where you give axioms for elementary euclidean geometry and then start to wonder wether the parallel axiom is necessary. You ... 0answers 41 views ### Geodesics of Sasaki metric I would like to ask the community for a reference on the following question: Let (M,g) be a Riemannian manifold and (T^1M,g_S) be the unit tangent bundle with the Sasaki metric. Is it true that ... 0answers 48 views ### Geodesics without a metric By definition, a geodesic is a mapping \gamma: I = (0, 1) \rightarrow M such that \nabla_{\dot{\gamma}(t)} \dot{\gamma}(t) = 0. Here we only need the connection. So, we do not need a metric to ... 0answers 118 views ### Exponential map on a sphere in spherical coordinates Let M = \{ (x^\varphi, x^\theta) : x^\varphi \in [0, \pi), \thinspace x^\theta \in [0, 2\pi) \} be a manifold with metric \mathrm{d}s^2 = (\sin x^\varphi)^2 (\mathrm{d} {x^\theta})^2 + (\mathrm{d} {... 1answer 45 views ### Reference on manifolds with boundary I am here because I want to know if someone knows of some good e fast books or references about manifolds with boundary. Help me please. 3answers 93 views ### Can we bypass connection? I am new to differential geometry. It is surprising to find that the linear connection is not a tensor, namely, not coordinate-independent. Can we bypass this ugly object? Only intrinsic quantities ... 2answers 398 views ### What hyperbolic space *really* looks like There are several models of hyperbolic space that are embedded in Euclidean space. For example, the following image depicts the Beltrami-Klein model of a hyperbolic plane: where geodesics are ... 0answers 262 views ### Advanced Differential Geometry Textbook In algebraic topology there are two canonical "advanced" textbooks that go quite far beyond the usual graduate courses. They are Switzer Algebraic Topology: Homology and Homotopy and Whitehead ... 1answer 37 views ### Is such kind of manifold Riemannian? Deforming the metric on the unit square by a weight applied in one direction If the metric is defined on a bounded subset of the x-y plane,let's say a closed square area 0\le x,y\le1 , the metric is defined as$$\langle u,v\rangle =\langle (u_x,u_y),(v_x,v_y)\rangle =\langle ... 1answer 129 views ### Solutions to Dirichlet problem on manifolds with boundary I am looking for a reference for the following assertion: Let$M$be a Riemannian manifold with boundary, and$f:\partial M \rightarrow \mathbb{R}$be smooth. Then there exists a unique smooth ... 0answers 50 views ### Parallel transform of a vector by Lie derivative I am new to differential geometry and I learn by myself. It seems that we need something extra called a connection to parallel transport a vector along a curve. But, suppose we have a vector field$...
My exact problem is a Riemannian manifold defined on $SU(2^n)$, where the metric is defined as follows: If $U(t)$ is a curve so that $U′(t)=−iH(t)U(t)$ (so just a unitary evolution of a quantum system ...