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### Comparison inequalities for Hamiltonian mechanics with convex potential - analogue to Rauch's theorem?

I'm asking about an area (Hamiltonian mechanics) that I don't know at all well; thus, I keep the question somewhat vague. In differential geometry, there are a number of results saying that geodesics ...
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### Why vanish the integer m of an ample line bundle in the Kodaira embedding theorem?

I try to understand the following version of the Kodaira embedding theorem: Let $X$ be a compact Kähler manifold. A line bundle $L$ is positiv if and only if it is ample. I have a problem with the '...
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### minimal surfaces in $S^n$

Thanks to Choi-Schoen theorem, we know that the space of embedded minimal surfaces into $S^3$ of fixed genus is compact. My question are simples: Can we remove the embeddness assumption? Can we ...
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### Example of steady Ricci soliton whith indefinite or nonpositive Ricci curvature

I am looking for example of steady Ricci soliton with indefinite or nonpositive Ricci curvature. Any help will be appreciated. Thanks!
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### Are there some intrinsic invariants of surfaces other than Gaussian curvature?

The principal curvatures of a surface is denoted by $\kappa_{1}, \kappa_{2}$. Let $P(x,y)$ be a polynomial with real coefficients. Assume that $P(\kappa_{1}, \kappa_{2})$ is an intrinsically ...
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### Techniques to solve a non linear differential equation related to curvature

Many years ago, I considered the following non linear differential equation: $y=y''.(1+y'^{2})^{-3/2}$ This equation expresses the equality between the value of a given function $y\in C^{2}(R)$ and ...
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### Regularization by mean curvature flow

I have a $C^{1,\alpha}$ surface defined as the graph of some function $\varphi : B \to \Bbb{R}_+$ ($B$ is a ball). This surface has positive and bounded mean curvature in the weak sense (since the ...
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### Hermitic connections on complex line bundles with imaginary curvature form

It is a simple fact that if $L \to B$ is a complex line bundle endowed with an Hermitian product and a compatible connection $\nabla$, then the curvature $F_\nabla$ is imaginary (and so are the local ...
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### Compressing a hypersurface on the sphere

Let $M^n$ be a compact, connected, orientable hypersurface of the unit sphere $S^{n+1} \subset \mathbb{R}^{n+2}$. Suppose $M$ is contained in the northern hemisphere $S_+^{n+1}$ and has nonzero ...
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### Differential geometric interpretation of cohomology

I'm not sure whether this is an appropriate question for this forum. I'm afraid that this is not a research level question however: 1. It's about reference request therefore the answer does not ...
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### On the curvature tensor with certain conditions

Let $(M^{n+m},g)$ be a Riemannian manifold and let $\lbrace X_1,...,X_n,Y_1,...,Y_m\rbrace$ be a locally orthonormal frame for $M$($3\leq n,m$). If we suppose the curvature tensor $R$ of $g$ ...
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### Regularity - mean curvature equation

In my research I arrived at the following equation: $$\int_B \frac{\nabla u \cdot \nabla \varphi}{ \sqrt{1+|\nabla u|^2}}=\int_B f \varphi, (*)$$ for every $\varphi \in C^1(B)$, which is a weak form ...
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### Holonomy of a Ricci-flat affine connection

There is some link between Ricci-flatness and reduction of holonomy. For example a Kahler manifold is Ricci-flat if and only if it has at most $SU(n)$ holonomy rather than $U(n)$, and it's apparently ...
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### Flows associated with Killing fields

Let $M$ be a Riemann manifold and $p, q$ two points on a geodesic $\sigma$ which are isotropically conjugate. That is, there is a Jacobi field along $\sigma$ vanishing at $p$ and $q$ which is the ...
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### A Converse to Cartan–Hadamard theorem?

Let $M$ be a complete Riemannian manifold, with the property that $\exp_p\colon T_pM \to M$ is a diffeomorphism for every $p \in M$. Can we say something about it's curvature? Is it true that its ...
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Is the following true? I cannot see a counterexample and it seems very intuitively clear, at least in the embedded case. Claim: Consider the set $S$ of closed immersed Riemann surfaces $\Sigma \... 0answers 38 views ### On a conjecture about Riemannian metric with positive sectional curvature [duplicate] What is the last status of the following conjecture? Is it still open? What partial or similar results are known up to now? Conjecture:$S^2\times S^2$admits a Riemannian metric with positive ... 1answer 231 views ### What is known about Lie groups with positive(strictly) curvature? If we consider$G$a Lie group with left invariant riemannian metric its sectional curvature is nonnegative, when this metric is positive? I thought a little about and only found$SU(2)=S³$. In ... 0answers 156 views ### Obtaining the metric from the mixed Ricci tensor$R^i{}_j$In chapter 5 of the book "Einstein Manifolds", Arthur Besse discusses the possibility to find the metric$g$when knowing the Ricci curvature tensor$Ric(g)$($=R_{ij}$). But what do we know about ... 1answer 99 views ### Prescribing an induced metric We know that, if we have a surface$z=f(x,y)with Euclidean space being ambient manifold, the induced metric is as follows (in matrix form): g=\begin{bmatrix} 1+\left ( \frac{\partial f(x,y)}{\... 0answers 76 views ### Classification of manifolds with {\rm Ric}\geq 0 wrt fundamental group Note that n-manifolds M with {\rm Ric}\geq 0 has a fundamental group of polynomial growth of degree \leq n (proof : use Bishop volume theorem). (Here a group \Gamma is said to have ... 1answer 91 views ### Convex embedding with a positivity condition We have a n-dimensional hypersurface \Sigma embedded in the Euclidean (n+1)-space \mathbb{R}^{n+1}. We know that \Sigma is compact without boundary, convex (not necessarly strictly convex), ... 2answers 484 views ### Calculating the Riemann Christoffel tensor for a diagonal metric I am trying to calculate the entries of the Riemann curvature tensor R^m_{\phantom{m}ijk} for the metric g_{ij}. The Riemann-Christoffel tensor is given as \begin{align} R^m_{\phantom{m}ijk} = \... 1answer 129 views ### The Laplacian of an expression involving the Ricci tensor While doing some computations on a compact Riemannian manifold I have reached the following expression: \Delta_y \big( Ric_y (\exp_y ^{-1} x, \exp_y ^{-1} x) \big) (x)$$where \Delta_y is the ... 2answers 752 views ### Square of the distance function on a Riemannian manifold Let (M^n,g) be a smooth Riemannian manifold. Consider the square of the distance function$$dist^2\colon M\times M\to \mathbb{R}$$given by (x,y)\mapsto dist^2(x,y). It is easy to see that this ... 0answers 45 views ### Normal fields of geodesic spheres This question is related to this one (http://math.stackexchange.com/questions/1383511/normal-curvature-of-geodesic-spheres) I've asked at math.stackexchange. Let (M,g) be a compact Riemannian ... 1answer 119 views ### Limited expansion of mean curvature of geodesic spheres I am working with the Laplacian on a Riemannian manifold (M,g) (compact, without boundary). In spherical geodesic coordinates (r, \sigma) around some arbitrary x \in M (where \sigma denotes ... 1answer 187 views ### Symmetries of non-Riemannian curvature tensor The curvature tensor, R_{ab}{}^c{}_d, can be obtained from a connection which not necessarily is a metric connection. By construction it is antisymmetric in the first two indices, since roughly ... 1answer 138 views ### The sign of the mean curvature on convex cones in three dimensions My question is as follows: It is known that a closed smooth curve in \mathbb{R}^2 is convex iff its (signed) curvature has a constant sign. I wonder if one can characterize smooth convex cones in \... 0answers 80 views ### Bounding distance between geodesics in manifolds with nonpositive curvature This is a duplicate of a question at the stackexchange which was not answered. I've recently read (in some notes by Mark Pollicott) the following related claims, which, although quite intuitive, I ... 0answers 75 views ### Curvature tensor for a singular manifold Given a manifold M with its tangent space TM and frame vector field e \in TM. However, the transition functions in this tangent bundle are non-smooth. Therefore, the Lie derivative of e with ... 0answers 73 views ### What is the relation between two Riemannian metrics with the same Riemannian curvatures? Consider two flat Riemannian metrics on a manifold. What is the general relation between these two metrics if the manifold is not simply-connected? What is the answer if two Riemannian metrics have ... 0answers 179 views ### Spherical cap is the only compact constant mean curvature surface bounded by a circle I would like to see that the only compact rotationally invariant constant mean curvature surfaces with boundary a planar circle, are either a planar disk or a spherical cap. This is stated in the ... 0answers 80 views ### Can a cylinder be regarded as a Riemannian manifold? [closed] Consider the surface of a bounded cylinder consisting of a top,bottom and side part together with the metric induced by the euclidean norm on \mathbb{R}^3. Can this space be regarded as a Riemannian ... 0answers 99 views ### Space with Ric \geq -(n-1) Note that hyperbolic space H has Ric=-(n-1). I want to know : Question : Does there exists a simply connected open complete Riemannian manifold M s.t. (1) Ric\geq -(n-1) on M (2) ... 1answer 158 views ### Ricci Curvature and the Chern Class of the Levi-Civita For a (compact) Kahler manifold M, the Ricci tensor is the symmetric 2-form$$ r(u,v) = \text{tr}\big( w \mapsto (D_wD_u - D_uD_w - D_{[u,w]})v\big). $$The Ricci curvature is the 2-form$$ r(... 2answers 220 views ### Projective curves of constant curvature A nodal projective curve in\mathbb{CP}^2$inherits a Kähler metric from the Fubini-Study metric, and hence a Riemannian metric. In particular, with respect to this metric, a line has constant ... 2answers 2k views ### Unifying Geometry for Characteristic Classes When working with characteristic classes (more concretely Chern classes), one finds at least four essentially distinct approaches: Axiomatic Approach. See, for instance, Vector Bundles and K-Theory, ... 1answer 181 views ### Gaussian Curvature of Exponentiated 2-Planes Consider a Riemannian manifold$M$with sectional curvatures$K\ge 0$and let$\Pi$be a 2-plane in the tangent space of$M$at a point$p$. In a small enough neighborhood$U$of 0 the exponential map ... 5answers 594 views ### Variation of curvature with respect to immersion? Let$M$be a smooth surface and let$f: M \to \mathbb{R}^3$be a family of immersions given by $$f(t) = f_0 + tuN_0,$$ where$f_0$is some initial immersion,$N_0$is the associated Gauss map, and ... 1answer 169 views ### Vertices of Curves and Eigenvectors of Hessian This might be a trivial question, but I can't seem to figure it out. Suppose I have an implicitly defined curve in the plane given by$f(x,y) = t$. This curve is strictly convex, and feel free to ... 0answers 91 views ### Exposition of the Calabi complex I am interested in a complex derived by Eugenio Calabi in his article "On compact Riemannian manifolds with constant curvature". The complex is referenced as "Calabi complex" in various citing ... 5answers 2k views ### Can anyone give an example of Ricci flat Riemannian or Lorentzian Manifold that is not flat? Does there exist a Ricci flat Riemannian or Lorentzian manifold which is geodesic complete but not flat? And is there any theorm about Ricci-flat but not flat? I am especially interset in the case ... 1answer 334 views ### curvature and volume growth Let$M$be a non-compact connected Riemannian manifold with$\mathrm{sec}_g=0$and$\operatorname{vol} B(x,r)\geq c(n)r^n$for any$r$, where$c(n)>0$. How to prove that$(M,g)$is isometric to$(R^...
Given a compact connected Riemannian $2$-manifold $M$ with positive curvature (thus by Gauß-Bonnet, $M$ is diffeomorphic to a 2-sphere) and diameter 1, what is the supremum (as $M$ varies over all ...
Let us assume $\text{H}$ is the mean curvature of a compact surface in $\mathbb E^3$ and $g$ is its genus. When $g$ is arbitrary, we have $\int_{\mathbb S^2}\text{H}^2dV=4\pi$ and \$\int_{\Sigma}\...