Integral formulas in riemannian geometry
01 Mar 1973-Bulletin of The London Mathematical Society (Oxford University Press (OUP))-Vol. 5, Iss: 1, pp 124-125
About: This article is published in Bulletin of The London Mathematical Society.The article was published on 1973-03-01. It has received 331 citations till now. The article focuses on the topics: Riemannian geometry & Fundamental theorem of Riemannian geometry.
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TL;DR: In this article, it was shown that the commutator of two soliton vector fields with the same metric in a given conformal class produces a Killing vector field, and that the corresponding vector field becomes a geodesic vector field if and only if the manifold is of constant curvature.
Abstract: In this paper we have obtained evolution of some geometric quantities on a compact Riemannian manifold $M^n$ when the metric is a Yamabe soliton. Using these quantities we have obtained bound on the soliton constant. We have proved that the commutator of two soliton vector fields with the same metric in a given conformal class produces a Killing vector field. Also it is shown that the soliton vector field becomes a geodesic vector field if and only if the manifold is of constant curvature.
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TL;DR: In this article, a new Hardy type inequality was established via a new cohomology vanishing theorems for free boundary compact submanifolds with $n\geq2$ immersed in the Euclidean unit ball.
Abstract: In this paper, via a new Hardy type inequality, we establish some cohomology vanishing theorems for free boundary compact submanifolds $M^n$ with $n\geq2$ immersed in the Euclidean unit ball $\mathbb{B}^{n+k}$ under one of the pinching conditions $|\Phi|^2\leq C$, $|A|^2\leq \widetilde{C}$, or $|\Phi|\leq R(p,|H|)$, where $A$ $(\Phi)$ is the (traceless) second fundamental form, $H$ is the mean curvature, $C,\widetilde{C}$ are positive constants and $R(p,|H|)$ is a positive function. In particular, we remove the condition on the flatness of the normal bundle, solving the first question, and partially answer the second question on optimal pinching constants proposed by Cavalcante, Mendes and Vitorio.
TL;DR: In this article, it was shown that the critical points of the energy functional of a vector field of a Riemannian manifold are critical points on the space of all vector fields of the manifold.
Abstract: Given a compact Lie subgroup G of the isometry group of a compact Riemannian manifold M with a Riemannian connection $$
abla ,$$
a G-symmetrization process of a vector field of M is introduced and it is proved that the critical points of the energy functional $$\begin{aligned} F(X):=\frac{\int _{M}\left\|
abla X\right\| ^{2}\mathrm{d}M}{\int _{M}\left\| X\right\| ^{2}\mathrm{d}M} \end{aligned}$$
on the space of $$\ G$$
-invariant vector fields are critical points of F on the space of all vector fields of M and that this inclusion may be strict in general. One proves that the infimum of F on $${\mathbb {S}}^{3}$$
is not assumed by a $${\mathbb {S}}^{3}$$
-invariant vector field. It is proved that the infimum of F on a sphere $${\mathbb {S}}^{n},$$
$$n\ge 2,$$
of radius 1 / k, is $$k^{2},$$
and is assumed by a vector field invariant by the isotropy subgroup of the isometry group of $${\mathbb {S}}^{n}$$
at any given point of $${\mathbb {S}} ^{n}$$
. It is proved that if G is a compact Lie subgroup of the isometry group of a compact rank 1 symmetric space M which leaves pointwise fixed a totally geodesic submanifold of dimension bigger than or equal to 1, then the infimum of F is assumed by a G-invariant vector field.
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TL;DR: In this paper, the authors consider almost Ricci-Yamabe soliton in the context of certain contact metric manifolds and show that the potential vector field is a constant multiple of the Reeb vector field.
Abstract: We consider almost Ricci-Yamabe soliton in the context of certain contact metric manifolds. Firstly, we prove that if the metric $g$ admits an almost $(\alpha,\beta)$-Ricci-Yamabe soliton with $\alpha
eq 0$ and potential vector field collinear with the Reeb vector field $\xi$ on a complete contact metric manifold with the Reeb vector field $\xi$ as an eigenvector of the Ricci operator, then the manifold is compact Einstein Sasakian and the potential vector field is a constant multiple of the Reeb vector field $\xi$. Next, if complete $K$-contact manifold admits gradient Ricci-Yamabe soliton with $\alpha
eq 0$, then it is compact Sasakian and isometric to unit sphere $S^{2n+1}$. Finally, gradient almost Ricci-Yamabe soliton with $\alpha
eq 0$ in non-Sasakian $(k,\mu)$-contact metric manifold is assumed and found that $M^3$ is flat and for $n>1$, $M$ is locally isometric to $E^{n+1}\times S^n(4)$ and the soliton vector field is tangential to the Euclidean factor $E^{n+1}$. An illustrative example is given to support the obtained result.
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TL;DR: In this article, the authors connect the above-mentioned descriptions using the notion of a harmonic vector field on the upper half plane of a compact Riemann surface, which takes inspiration from the theory of harmonic maps between compact hyperbolic surfaces.
Abstract: Usually, the description of tangent spaces to the Teichmueller space $\mathscr{T}(\Sigma_{g})$ of a compact Riemann surface $\Sigma_{g}$ of genus $g \geq 2$ (which we can identify with the quotient space $\mathbb{H}^{2} / \Gamma_{g}$ of the upper half plane $\mathbb{H}^{2}$ by a discrete cocompact subgroup $\Gamma_{g}$ of $\mathrm{PSL}(2, \mathbb{R})$) comes in two different flavours: the space of holomorphic quadratic differentials on $\Sigma_{g}$ which are holomorphic sections of the tensor square of the canonical line bundle of $\Sigma_{g}$ and the first cohomology group $H^{1}(\Gamma_{g}; \mathfrak{g})$ of the fundamental group $\Gamma_{g}$ of $\Sigma_{g}$ with coefficients in the vector space $\mathfrak{g}$ of Killing vector fields on $\mathbb{H}^{2}$ (or on $\mathbb{D}$), a.k.a the Lie algebra of $\mathrm{PSL}(2, \mathbb{R})$. In this article, we are concerned with connecting the above-mentioned descriptions using the notion of a harmonic vector field on the upper half plane $\mathbb{H}^{2}$ (equivalently, on $\mathbb{D}$) that takes inspiration from the theory of harmonic maps between compact hyperbolic Riemann surfaces. As an application, we also show that how a harmonic vector field on $\mathbb{H}^{2}$ (or on $\mathbb{D}$) describes a connection on the universal Teichmueller curve.