# Tagged Questions

Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.

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### Smooth morse theory of Riemannian distance functions

Let $(M,g)$ be a Riemannian manifold, and $p\in M$. As $R>0$ increases, the topology of the ball $B(p,R)$ changes, but the changes happen only at a Lebesgue measure zero set of $R$. For instance, ...
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### Decomposition of a closed surface

I know that I can decompose an hyperbolic closed surface of genus $g>1$ into $2(g−1)$ pants bounded by $3$ geodesics. It seems reasonable to think the same can be done for a closed surface of genus ...
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### Do smooth manifolds create colimits for complex manifolds?

Suppose we have a diagram $D$ in the category $\textrm{Diff}_\mathbb{C}$ of complex manifolds, and suppose this diagram has a colimit $L$ after inclusion in the category $\textrm{Diff}$ of smooth ...
338 views

### Monge–Ampère with drift

Let $I\subseteq \mathbb{R}$ be an interval. Let smooth $M(x,y):I\times(0,\infty) \to \mathbb{R}$ satisfies PDE: $$M_{xx}M_{yy}-M_{xy}^{2}+\frac{M_{y}M_{yy}}{y}=0.$$ My question is to ...
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### Volume-minimizing submanifold implies calibrated?

Let $X$ be a smooth manifold of dimension $d$ and $M$ an oriented submanifold of dimension $p < d$ so that the multiples k⋅M are absolutely minimizing $p$-volume in their integral homology ...
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### Marcel Berger's “Sur les groupes d'holonomie homogènes de variétés à connexion affine et des variétés riemanniennes.”

I would appreciate any reference that contains either a translation or proof of the main theorem in this paper. Thank you in advanced.
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### 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 ...
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### Volume element of symmetric definite matrices in polar coordinates

I have a difficulty to understand the following statement. I don't ask for a proof but just understand the statement concretely (what it does mean, how to apply it...) Let $\mathcal P_n$ be the ...
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### Taylor expansions of Riemannian exponential map and Jacobi fields? [closed]

Apologies if this is not exactly a research-level questions, but I've no known reference where I can figure it out myself. I asked this on math.stackexchange.com, ...
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### Positive solutions to Yamabe problem?

Yamabe problem ensures that for any Riemannian metric $g$, in its conformal class $[g]$ there always exists a metric $\bar g$ whose scalar curvature $\bar R$ is constant. I was wondering whether ...
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### Spherical harmonics and ellipticity of the Laplacian

Let us consider the sphere $S^n$ and the Laplacian $-\Delta$ on it. Let $L^2(S^n) = \bigoplus_k V_k$, where $V_k$ represents the eigenspace of the Laplacian with eigenvalue $k(k + n - 1)$. We know ...
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### Reference request: Intrinsic definition of the strong Whitney topology on $\mathcal{C}^{\infty}(M,\mathbb{R})$ without using charts or jets

Let $M$ and $N$ be smooth manifolds. There is a description of the strong Whitney topology on $\mathcal{C}^{\infty}(M,N)$ in terms of partial derivative in charts (using locally finite sets of charts ...
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### Differential and pre-differential of Jacobi identity

Let M be a manifold. To what extent all Lie algebra structures with tensorial property on $\chi^{\infty}(M)$ are studied? That is a Lie algebra structure for which $[X,fY]=f[X,Y]$. (For ...
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### existence of totally geodesic hypersurfaces

Assume we are on a smooth, complete Riemannian manifold $(M,g), dim(M) \geq 3$. What are the specific geometric/topological constraints for such a manifold to admit complete, totally geodesic ...
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### Kerr metric affine parameter

I'm going through the chapter about Kerr space-time of Chandrasekhar's "Mathematical theory of black holes", and have a question about the following transformation: the idea is, that one wants to ...
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### Are maps homotopic with respect to a uniform number of local homotopies

I've encountered the following problem that I'm sure someone more topologically inclined can answer: Say that a homotopy of maps $f:X\times[0,1)\to Y$ between two compact smooth manifolds $X$ and $Y$ ...
Let $M,N,X$ be smooth manifolds. Equip the space of smooth functions between two manifolds with the (strong) Whitney- $\mathcal{C}^\infty$-topology. The evaluation map $$ev\colon ... 0answers 118 views ### Lie derivative and taking trace Let (M,\omega) be a complex Kahler manifold, and g is a smooth function such that \int_Mg\omega^n=0. It is obvious that there exists a smooth function f such that \triangle_\omega f=g. ... 0answers 98 views ### Certain principal bundle structure on \mathbb{R}^{n} \setminus \{0\} I ask this question in MSE and I received no answer, so I repeat it here: Is there a right action of \mathbb{H}^{2} on some \mathbb{R}^{n}\setminus \{0\} such that this action gives us a ... 1answer 127 views ### derivative of the adiabatic limit of the eta invariant To ask my question I have to write down the setup. Basically the setup is the adiabatic limit of the reduced eta invariant of Dirac operator associated to the submersion metric and connection. So if ... 0answers 70 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 111 views ### Moduli space of line-bundle holomorphic structures and sections over a Kahler-Hodge manifold Let (M,\omega) be a Kahler manifold with Kahler integral two-form \omega and let (L,h) be a rank-one complex vector bundle over M equipped with a fixed hermitian metric h. I am interested in ... 0answers 361 views ### Vafa's semi-Ricci flat metric Cumrun Vafa with Greene-Shapere-Yau introduced semi-Ricci flat metric here B. Greene, A. Shapere, C. Vafa, and S.-T. Yau. Stringy cosmic strings and noncompact Calabi-Yau manifolds. Nuclear Physics ... 1answer 133 views ### Extension of a smooth function from a convex set Let C\subset \mathbb{R}^{n}, C'\subset\mathbb{R}^{m} be two convex sets with a non-empty interior. A function F\: : \: C\to C' is said to be differentiable at x\in C if there exists a linear ... 0answers 80 views ### Differential form heat kernel on hyperbolic space Is there an explicit formula in the literature for the heat kernel of the Hodge Laplacian on differential forms? I found some on functions, but not on forms of higher degree. What at least about ... 2answers 449 views ### Who is currently researching topics concerning applying algebraic topology and/or differential geometry to numerical methods? [closed] I am interested in pursuing a PhD in mathematics from a top ranked university with a faculty member researching something akin to the following description: applications of algebraic topology and/or ... 2answers 129 views ### Differentiability of polytope shadow areas Let P be an opaque convex polyhedron containing the origin in \mathbb{R}^3, and let S be an origin-centered sphere strictly containing P: S \supset P. For a point x on S, let \sigma(x) ... 1answer 153 views ### Generalisation of “tangent space” to not-necessarily connected sets I vaguely recall having read somewhere a definition similar to (but probably not exactly the same as) the following. Definition (Blob) Let S\subset \mathbb{R}^n be a set, and p \in S. The ... 0answers 21 views ### Restrictions of potential tensor fields to toric subgroups Let G be a compact connected nonabelian Lie group and let f be a symmetric tensor field of order m\geq1 on G. Let T\subset G be a translate of a torus subgroup of G with \dim(T)\geq1. ... 0answers 168 views ### Kähler differentials, intuition behind \text{div}(\omega), canonical divisor on algebraic curves? See my two previous questions here: Intuition for thinking about R-module of Kähler differentials, universal receptacles, derivations? and Kähler differentials, define valuation? for background. If ... 0answers 106 views ### Metric(s) on Grassmann Manifold and Plucker Embedding I'm working on a numerical optimization problem that naturally lives on the Grassmann Manifold Gr_N(\mathbb{C^M}), however the objective function is defined on the alternating algebra given by the ... 1answer 185 views ### Pontryagin Forms and Special Holonomy Let (M,g) be a Riemannian manifold. Recall that the k^{th}-Pontryagin class is a topological invariant which, by classical Chern-Weil theory, can be represented using the so-called Pontryagin ... 1answer 104 views ### Examples of Smooth, Compact and Non-rigid Manifolds that Bound a Finite, Non-zero Volume Are there codimension-1 submanifolds of \mathbb{R}^n, that are smooth everywhere and topological equivalent to a sphere with 0\le h\lt\infty handles and allow an isometric deformation, that ... 1answer 196 views ### Intuition for thinking about R-module of Kähler differentials, universal receptacles, derivations? Suppose k is a field of characteristic zero, and R is a k-algebra. The R-module of Kähler differentials \Omega_{R/k} of R over k with generators \{dr\}_{r \in R} is the module subject ... 0answers 76 views ### Gradient vector fields defined with respect to two different metrics and Morse theory Given a differentiable manifold M, we can equip M with a Riemannian metric g or g' to generate a pair of Riemannian manifolds (M,g) and (M,g'), respectively. The gradient vector fields ... 0answers 89 views ### Mapping theorem in higher dimensions The Riemann mapping theorem states that given any two simply connected open domains A and B of \mathbb C that are neither empty nor equal to \mathbb C, there exists a unique (up to ... 1answer 104 views ### Norm equivalent to Sobolev norm? [closed] On the hyperbolic space \mathbb{H}^n, it is known that the spectrum of the Laplacian satisfies \text{Spec}(-\Delta) \subset [\frac{(n - 1)^2}{4}, \infty). Consider the operator P = -\Delta + a, ... 1answer 150 views ### when are local quasigeodesics global in CAT(0) It is well-known (and easily shown) that a local quasi-geodesic (for some value of "local") in a \delta-hyperbolic space is global (one can compute the constants, as well, from local data). This is ... 0answers 120 views ### What is the symplectic manifold whose Delzant polytope is a trapezoid? What is the symplectic form on the manifold whose associated Delzant polytope is a trapezoid? I am trying to find it by using the Marsden–Weinstein theorem, but I have been unable to do so. If ... 1answer 231 views ### The existence of differential operator of the form AB=0 We define \mathcal A is a differential operator of order n with variable coefficients if$$ \mathcal A:=\sum_{|\alpha|\leq n}A_\alpha (x) D^\alpha $$where \alpha is an muti-index and ... 2answers 181 views ### Totally geodesic submanifold of a hyperbolic 3-manifold If M is a convex-cocompact hyperbolic 3-manifold, and S is a closed surface with genus \geq 2. Suppose f:S\to M is a minimal immersion, and f(S) is negatively curved. I know that all the ... 1answer 144 views ### Constant spinors from constant forms Let (X,g) be a m-dimensional complex, hermitian, spin manifold and let us denote by S_{\mathbb{C}} its complex spinor bundle. Then: S_{\mathbb{C}}\simeq \Lambda_{\mathbb{C}}(X) Let \nabla ... 0answers 89 views ### Connection and reduction of the structure group I apologize for this question which is not really research-level, but I don't get any answer on mathStackExchange, and I asked a professor in my university... which couldn't find the solution. I am ... 1answer 148 views ### Using Jacobi fields to approximate parallel transport along geodesic:is the following limit true? I apologize if this is not a research level question (already tried asking http://math.stackexchange.com/questions/1303288/relation-between-parallel-transport-and-jacobi-field-iion stack exchange with ... 2answers 154 views ### Special connection of vector bundle over real manifold Let E \rightarrow M be a vector bundle over a smooth manifold M and let g be a bundle metric. Does there exists a conection (maybe unique) \nabla which is compatible with g. By this I mean: ... 0answers 119 views ### There is no quasiregular diffeomorphism from punctured ball into ring (on the plane) The idea is to use l2 cohomology as a quasiregular map invariant. It is easy to see that there are closed 1-forms on the ring which are not exact, but it occures that every closed l2-form f_1(x,y)dx ... 2answers 296 views ### Lower bound on the first eigenvalue of the Laplace-Beltrami on a closed Riemannian manifold It has been proved by Li-Yau and Zhong-Yang that if M is a closed Riemannian manifold of dimension n with nonnegative Ricci curvature, then the first nonzero eigenvalue \lambda_1(M) of the ... 1answer 226 views ### How many Fréchet manifolds are there? Clearly the title needs clarifying. Allow me to let "how many" to mean a set larger than a skeleton of the category of Fréchet manifolds and smooth maps, if this category is indeed essentially small. ... 3answers 369 views ### A \wedge A \wedge A in Chern-Simons I am confused with the wedging operations of Lie algebra valued differential forms. Especially, for instance, I have some problems with the Chern-Simons 3-form$$A \wedge dA + \frac{2}{3}A \wedge A ...
Let $M$ be compact $G$-homogeneous manifold, equipped with the equivariant complex structure, when $G$ is a semi-simple algebraic group. The obvious example is every flag manifold. In that case, all ...