Questions tagged [riemannian-geometry]
Riemannian Geometry is a subfield of Differential Geometry, which specifically studies "Riemannian Manifolds", manifolds with "Riemannian Metrics", which means that they are equipped with continuous inner products.
15 questions from the last 30 days
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Question on gamma matrices
Let $(M,g)$ be a pseudo-Riemannian spin manifold and let us denote by $S$ the spinor bundle, i.e. the associated vector bundle with respect to the spin representation. Usually, the "gamma ...
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Request for resources on directional derivative of the Riemannian distance function, and Berger's lemma about geodesics realizing the diameter
I've been recently interested in directional derivatives of the Riemannian distance function, and I came across this question, and its answer by Sergei Ivanov, where he stated an important result: (I ...
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What do we know about Poisson boundaries of arbitrary Riemannian manifolds?
For closed manifolds, we know that the Poisson boundary is trivial due to compactness and for radially symmetric manifolds for which diffusion is one dimensional, there are A Brief Introduction to ...
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Harmonic map in the homotopy class of the identity map
Eells and Sampson's existence Theorem states that if $(N, h)$ is nonpositively curved, then a given map $f : (M, h') \to (N, h)$ can be deformed into a harmonic map in its homotopy class. Here smooth ...
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Connectedness of the space of negatively curved metrics of a compact 3-manifold
Is the space of metrics of negative sectional curvature over a closed 3-manifold connected? If so, in what paper is this result stated?
Note: as the Ricci flow hyperbolizes negatively curved metrics, ...
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A question in spin geometry in dimension 8
$\DeclareMathOperator\trace{trace}\DeclareMathOperator\End{End}\DeclareMathOperator\Trace{Trace}$This is to understand a very specific isomorphism in dimension $8$. In dimension $4$ for a spin$^c$ ...
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Reference request: $\operatorname{Sym}^2_0(T^*M) \simeq \Lambda_- \otimes \Lambda_+$
I am looking for proof of the "well-known" result that for a $4$-dimensional Riemannian manifold $(M, g)$, we have an isomorphism
$$
\operatorname{Sym}^2_0(T^*M) \simeq \Lambda_- \otimes \...
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Example of non homogenous manifold with a finitely generated algebra of natural functions
Let $(M,g)$ be a Riemannian manifold. Let $C^{\infty}_{Nat}(M,g)$ be the $\mathbb{R}$-algebra of scalar invariants of the curvature tensor and all its higher covariant derivatives. An example of a ...
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Calculi of pseudodifferential operators and K-theory
I am reading the thesis of Chris Kottke (https://dspace.mit.edu/bitstream/handle/1721.1/60193/681923895-MIT.pdf) and I would need some help to try to understand intuitively why he makes the choice of ...
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Question on Lorentzian geometry
I apologize in advance if this is a too basic question.
Let $(M,g)$ be a Lorentzian manifold with signature convention $(-,+,\dots,+)$. Now, lets suppose $X\in\Gamma(TM)$ defines a global time-...
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Regularity of the eigenfunctions associated to perturbed laplacian on a compact manifold
Let $M$ be a closed manifold, I consider first order laplacian perturbation associated to a density $\rho \in \mathcal{C}^\infty(M)$ with $\rho > 0$ of the form :
$$
\Delta_{\rho} f = \Delta f + \...
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Existence of Kähler Metric of Bounded Geometry on the Hermitian Vector Bundle on Projective Spaces
A Riemannian manifold $(M,g)$ is said to be of bounded geometry if the Riemannian curvature tensor and its derivatives are bounded, and it has positive injectivity radius.
I am working with the ...
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Geodesic flows and Killing fields
How well-known is the following result:
Let $(M,g)$ be a closed Riemannian manifold and define $D:C^\infty(M,Sym^mT^*M)\to C^\infty(M,Sym^{m+1}T^*M)$, $D\alpha:=\mathbb S(\nabla\alpha)$ where $Sym^m$ ...
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Comparison of special metrics on Riemann surfaces with the hyperbolic one
Let $X$ be a complex algebraic curve, or equivalently a compact Riemann surface, of genus $a\ge2$. Denote $\omega_1, \dots , \omega_a$ a basis of holomorphic differentials, and consider the Riemaniann ...
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Tilings of $\mathbb{R}^n$ and Riemannian manifold that is uniformly locally isometric to a ball in $\mathbb{R}^n$
Suppose that we have a Riemannian manifold $(M, g)$ that is uniformly locally isometric to a ball in $\mathbb{R}^n$, that is, there exists $r > 0$ such that for every $x \in M$ ball $B(x,r)$ in $M$ ...
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