All Questions
Tagged with reference-request dg.differential-geometry
800 questions
2
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381
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Question about Berezin line bundle of odd cotangent bundle of supermanifold $\text{Ber}(\Pi T^*\mathcal N)$
The followings are from Mnev's paper about BV formalism.
Example 4.15 (Definition of split supermanifold)
Let $E \to M$ be a rank $m$ vector bundle over $n$-manifold $M$, then there exists a ...
- 79
2
votes
1
answer
210
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CW-structure induced by Morse function on Riemannian manifold [duplicate]
I have heard a statement in the following direction. Given a compact Riemannian manifold $M$ with a Morse function on it possibly satisfying some extra assumptions. Then this data induces a CW-...
- 21.8k
2
votes
1
answer
149
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Surveys/monographs on the vortex filament equation
Where can I find surveys on the mathematical aspects of the vortex filament equation?
In particular, I'm interested in the following topics:
physical motivation;
notion of solutions and ...
- 277
2
votes
1
answer
214
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What is the space parametrising the curved sub-Cartan geometries of a flat Cartan geometry?
I'm basically wondering how to make "curved" the first column of the diagram
$\require{AMScd}$
\begin{CD}
P_1 @>\textrm{inclusion} >> G\\
@V \omega_0 V P_1\cap P_2 V @V\omega V P_2 V\\...
- 2,527
2
votes
2
answers
415
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Geometrical interpretation of a Schrödinger operator
Consider a $2 \times 2$ Hermitian (or symmetric) matrix-valued function
$$g(x) = \{ g_{jk}(x)\}_{j,k=1,2}, \quad x \in \mathbb{R}^{2},$$
such that $0 < m_{-}I \leq g(x) \leq m_{+}I$, for some $m_{-}...
- 321
2
votes
1
answer
255
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Parameter dependent differential equation in a Lie group
It is well-known that a linear differential equation in a finite-dimensional vector space depends continuously on some external parameters (for details see below). I search for an explicit reference ...
- 5,824
2
votes
2
answers
519
views
How to use that the Hessian is negative definite in this proof
Let $X$ be a Riemannian compact manifold on which acts a compact Lie group $H^+$. Let $f^+ : X \rightarrow \mathbb{R} $ be a smooth function on $X$. Consider a Lie subgroup $U$ of $H^+$ and suppose ...
- 139
2
votes
1
answer
224
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The differentiability of the distance function on asymptotically flat manifolds
Let $M = \mathbb{R}^3 \setminus \overline{B_1}$ where $\overline{B_1}$ is the closed unit ball.
Let $g$ be an asymptotically flat metric of the form $g_{ij} = \delta_{ij}+h_{ij}$ in standard ...
- 969
2
votes
2
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453
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Reference for the divergence theorem for embedded $C^1$-submanifolds of $\mathbb R^d$ with boundary
I'm aware of Gauss's theorem (aka the divergence theorem) for compact subsets $K$ of $\mathbb R^d$ with "$C^1$-boundary"$^1$.
I know that there are several generalizations of this theorem, ...
- 167
2
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1
answer
262
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A clarification regarding analytic perturbation of metrics and Laplacian
This question is in reference to the following Mathoverflow question and the accepted answer to it. It seems to me that it is taken for granted that if the metric $g_t$ perturbs real analytically in ...
- 123
2
votes
1
answer
164
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Curvature computations of globally symmetric spaces of rank $1$
I've been having trouble with finding the curvature computations of globally symmetric spaces of rank $1$.
More specifically, I need to use results about the eigenvalues of the operator $R:T_pM \...
- 401
2
votes
1
answer
83
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Calculating the Upper Bound on the Sphere Radius of Knotted Channel Surfaces
This question is motivated by trying to determine the upper bound on the thickness of a rope of fixed length (w.l.o.g. $2\pi$), with which a knot of given topology can be realized under the further ...
- 13.2k
2
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1
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333
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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 ...
- 1,407
2
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1
answer
490
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Curve on a surface defined by its geodesic curvature
Suppose that $S$ is a smooth complete surface, and $c\colon [0,L]\to S$ is a smooth curve in $S$, parametrized by arc-length. Then $c$ is uniquely determined by its initial tangent vector and its ...
2
votes
1
answer
391
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Fully non-linear PDE
A nice method of obtaining existence of solutions of many geometrically defined (and hence highly degenerate) parabolic systems (such as mean curvature flow) involves the reduction of the system to a ...
2
votes
1
answer
347
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Lee codes and $n$-torus
This is in continuation with this post:
Geometric/Analytic techniques for constructive and asymptotic bounds in the Lee metric
Codes over alphabet $\mathbb{Z}_{q}$ of length $n$ for the Lee metric ...
- 800
2
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1
answer
351
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On the Complete integrability of a tangent distribution
Reading about the geometrical theory of systems of first order pdes, I have met a result from symplectic geometry, that is easy to prove, but I am unable to give a reference for it. So my question is:...
- 4,306
2
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1
answer
239
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Projection of a ball in the ambient space to a manifold
Let $B_h (x)$ be the ball of radius $0<h \ll 1$ centered at $x\in \mathbb{R}^d$.
Let $I=[0,1]^{d-1}$ be the unit cube in $\mathbb{R}^{d-1}$, and let $f:I \to \mathbb{R}$ be a $C^2$ function. Then $...
- 3,574
2
votes
2
answers
332
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Heat kernel upper bounds on a complete Riemannian manifold
Let $M$ be a complete Riemannian manifold, and $p(t, x, y)$ denotes its heat kernel. I am trying to find sufficient conditions for when the following holds:
$$ p(t, x, y) \leq Ct^{-n/2}, \forall x, y, ...
- 21
2
votes
1
answer
217
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Regularity - mean curvature equation
In my research I arrived at the following equation:
$$ \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 ...
- 21
2
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0
answers
86
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Clarification about solvability of Dirichlet problem at infinity on a pinched negative curvature space
Let $M$ be a complete Riemannian manifold of pinched negative curvature $(-a^2 \leq K \leq -b^2 < 0)$. Let $M_\infty$ denote the ideal boundary and $\varphi \in C^0(M_\infty)$ be a prescribed "...
- 1,407
2
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0
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91
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Equivalence class of parametrized surfaces which induce the same current
Suppose $M$ is a smooth manifold of dimension $n \geq 2$. A $k$-current is a linear functional on compactly supported smooth forms on $M$, denoted $T: \Omega^k_c(M) \to \mathbb{R}$.
Let $X: [0,1]^2 \...
- 21
2
votes
0
answers
113
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What is known about warped product metrics satisfying conditions more general than conformal flatness?
In this paper, the authors characterize warped product metrics which are conformally flat (the fibers must have constant sectional curvature, on some cases there is a limitation on the number of ...
- 659
2
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0
answers
126
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Differential equations: trying to connect a nonlinear equation to a linear one
The following is motivated by taking a product space $\Omega$ and splitting it into two parts via projections, whose subspaces, $T$ and $X$, are home to functions which satisfy a nonlinear PDE and a ...
- 383
2
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0
answers
120
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The inverse of the metric on the space of harmonic p-forms
Let $X$ be a manifold equipped with a riemannian metric $g$. The space of harmonic p-forms is an $b_{p}(X)$ dimensional vector space with $b_{p}(X)$, the pth Betti number. One can then define a metric ...
- 141
2
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0
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251
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The conormal sheaf is the sheaf of sections of the conormal bundle for smooth manifolds
$\def\sO{\mathcal{O}}
\def\d{\mathrm{d}}$In ringed spaces theory, there is a notion of “conormal sheaf of an immersion” (mainly used in scheme theory), whereas in smooth manifold theory, there is the ...
2
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0
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132
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Elliptic equations and Fredholms alternative in the non-compact case
Let $M$ be a smooth Riemannian manifold and $E$ be a finite-rank vector bundle over $M$ equipped with a bundle metric $\langle\cdot,\cdot\rangle\in\Gamma^{\infty}(E^{\ast}\otimes E^{\ast})$, i.e. $\...
- 1,429
2
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0
answers
134
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Reference for Morse-Bott vector fields
I'm looking for a reference for the following result:
Let $X$ be a vector field on a manifold $M$ and let $C$ be a submanifold of $M$ formed by critical points of $X$. Assume that the Hessian of $X$ ...
- 1,409
2
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0
answers
109
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Further directions in representations of surface group into a Lie group
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PSL{PSL}$I studied the interpretation of Teichmuller space as a representation space for surface groups in $\PSL(2,\mathbb{R})$.
Now I am planning to ...
user498026
2
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0
answers
141
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Harmonic functions on varifolds
Let $T$ be a $k$-dimensional varifold in a Riemannian manifold $M$. Assume that $f$ is a smooth function on $M$ which is weakly (sub-)harmonic on $T$; that means that
$$
\int \langle \nabla_\omega f, ...
- 415
2
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0
answers
65
views
Coordinate free supersymmetric sigma model Lagrangian
I would like to know if there is a coordinate free version of the Lagrangian of the supersymmetric sigma model on a $2$-dimensional spacetime, with target space a Kähler manifold. The action for this ...
- 355
2
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0
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175
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Expository material on the Gromov-Lawson surgery theorem
I am looking for an expository text on the paper "The classification of simply connected
manifolds of positive scalar curvature" by Gromov and Lawson, in particular on the proof of Theorem A....
- 147
2
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0
answers
65
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Connection between a function and its usage in geometry [closed]
I know nothing about geometry, but I found a function which seems to have something to do with geometry.
This function is, $$f(x,y,z) = \dfrac{(x,y,z)}{\sqrt{1 + x^2 + y^2 + z^2}}$$
where $x,y,z$ is ...
- 121
2
votes
0
answers
664
views
Reference request - Texts on geometric analysis with exercises
I’ve recently been studying some Riemannian geometry and geometric analysis, however I have found it difficult to find resources with exercises to practice. It seems that many textbooks past the ...
2
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0
answers
71
views
Covariant momenta associated to higher order Lagrangians
Let $\pi:Y\rightarrow X$ be a fibered manifold with fibered coordinates $(U,x^i,y^\rho)$ (whenever local calculations are needed) and $m$ dimensional base $X$ ($\dim X=m$).
Suppose that $L\in\Omega^m_{...
- 2,792
2
votes
0
answers
127
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Are metrics of the form $dr^2+ \Omega^2 r^2 g_\text{round}$ asymptotically flat?
Let $M = [1,\infty)\times S^2$. Let $\Omega$ be any smooth positive function on $S^2$.
Is the metric $dr^2+ \Omega^2 r^2 g_\text{round}$ asymptotically flat (where $g_\text{round}$ is the round metric ...
- 969
2
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0
answers
65
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Where can I find a proof of the main properties of Weyl Curvature for semi-Riemannian manifolds?
Most of the references I've seen deal with Riemannian geometry, rather than semi-Riemannian geometry. Chens monograph, Pseudo-Riemannian Geometry, $\Delta$-Invariants and Applications is one of the ...
- 2,369
2
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0
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230
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Reference request: the UEA of the LR-algebra of tangent vector fields on a smooth manifold coincides with the derivation ring and the ring of diff ops
Let $\mathcal{M}$ be a smooth real manifold and let $A:= \mathcal{C}\left(\mathcal{M}\right)$ be the real algebra of smooth functions on $\mathcal{M}$.
Recall from McConnell, Robson, Noncommutative ...
- 718
2
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0
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116
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Access to an old paper of Obata
I'm trying to access the following paper of Morio Obata:
Conformal changes of Riemannian metrics on a Euclidean sphere, in "Differential Geometry -- In honor of K. Yano", 1972, pp. 347--353: ...
- 651
2
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0
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141
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For a 1-parameter family of metrics, how do we compute the derivative of the intrinsic geometrical objects like curvature, Hessian, etc
Consider a family of metrics and functions $(g(t), u(t))$ on $M:= \mathbb{R}^3 \setminus B_1$ satisfying
$$ g(0) = g_0, \quad g'(0) = \tilde g, \quad u(0) = u_0, \quad u'(0) = \tilde u$$
where $g_0$, $...
- 969
2
votes
0
answers
269
views
Solvability of a PDE involving the Dirichlet-to-Neumann operator
Let $M = \mathbb{R}^3 \setminus B_1$ where $B_1$ is the unit ball (equip $M$ with the euclidean metric for simplicity, but it will be replaced by an arbitrary asymptotically flat metric).
Let $N: L^2(\...
- 969
2
votes
0
answers
125
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Variation of Morse Functions: a reference request
Suppose I have a manifold $X$ and a family of Morse functions $F_t:X \times \mathbb R \to \mathbb R$ on it where $t$ is the second parameter. So, if we fix $t$, we get a regular Morse function for ...
- 7,746
2
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0
answers
152
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Period map on non-Kähler manifold
Is there a theory of period map on non-Kähler manifolds that has Hodge decomposition? Any reference is helpful. Thank you.
- 263
2
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0
answers
67
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Closed-form expression for Riemannian exponential maps on symmetric spaces
Besides the Poincaré model for the hyperbolic disc, $S^n$, and Euclidean space, what are known instances of a symmetric space $M$ (finite-dimensional) for which the exponential map is known in closed-...
- 5,405
2
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0
answers
172
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Construction of Kahler Einstein Metric of Poincare Type
I am reading Kobayashi's Kahler-Einstein metric on an open algebraic manifold. In this paper he constructs a Kahler-Einstein of Poincare type on an open manifold X' = X\D, where X is projective and D ...
- 31
2
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0
answers
134
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Hypersurfaces whose unit normal $N$ satisfies $[N,X] =0$ for every tangent vector field $X$
Let $M$ be a hypersurface of a Riemannian manifold, and assume that $M$ satisfies the following property:
For each $p \in M$, given a unit normal vector field $N$ defined in a neighborhood $U$ of $...
- 985
2
votes
0
answers
269
views
Extending Green's theorem from very special regions to more general regions
Green's theorem
Let $C$ be a positively oriented and consists of a finite union of disjoint,piecewise smooth simple closed curve in a plane, and let $D$ be the region bounded by $C$. If $P$ and $Q$ ...
- 151
2
votes
0
answers
255
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Sobolev Multiplication on non-compact manifold
We know that for a compact Riemannian $n$-dim manifold $(M,g)$(the boundary could be nonempty), the Sobolev Multiplication Theorem states that $L^p_k\times L^q_l⟶L^r_m$, where $1/r−m/n>1/p−k/m+1/...
- 1,915
2
votes
0
answers
66
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One-parameter group of nonvanishing vector field
Let $M$ be a smooth manifold( if necessary one can assume it is closed), $V$ be a non-vanishing vector field of $TM$.
Q: Under what condition, we can say that $\overline{\exp(tV)}$, i.e. the closure ...
- 1,915
2
votes
0
answers
126
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The complex Clifford algebra
If $(E,g,w)$ is a vector space $E$ with a metric $g$ and a symplectic form $w$; then we can define the complex parts $(1,0)$ and $(0,1)$, so that the complex Clifford algebra is: $$e_1 . f_1 + f_1 . ...
- 21