Weak Relatives: A Mathematical Perspective

• Weak Relatives are a relaxed notion where classical relations among structures are weakened, retaining local metric or analytic properties essential for classification and rigidity results.
• They appear in diverse areas such as Kähler geometry, locally compact groups, operator algebras, and Banach space theory, providing frameworks for embedding and fixed-point theorems.
• The concept enables nuanced distinctions like strict relatives and relative amenability, with implications for projective rigidity, fixed-point phenomena, and approximation properties.

Searching arXiv for papers on “weak relatives” and closely related usages to ground the article. arXiv search query: "weak relatives" In the available literature, “weak relatives” does not denote a single universal concept; it appears in several distinct technical settings in which an established relation is weakened while retaining enough structure to support classification, rigidity, or comparison theorems. The most explicit formal usage occurs in Kähler geometry, where two Kähler manifolds are weak relatives if they contain locally isometric Kähler submanifolds that holomorphically and isometrically immerse into the ambient manifolds. Closely related weakening phenomena also occur in locally compact groups, operator algebras, Banach operator ideals, and model theory, where the adjective “weak” typically marks a relaxation of a classical fixed-point, averaging, approximation, or extension property (Placini, 2023, Caprace et al., 2013, Lassalle et al., 2014, Malinowski et al., 2022).

1. Kähler-geometric weak relatives

Following Di Scala–Loi, two Kähler manifolds $M_1$ and $M_2$ are relatives if there exists a Kähler manifold $X$ and two holomorphic isometries $\varphi_i:X\to M_i$, $i=1,2$. The weakened notion replaces the common submanifold $X$ by two locally isometric Kähler manifolds $X_1$ and $X_2$ of complex dimension $\ge 2$, together with holomorphic isometries $\varphi_i:X_i\to M_i$. Thus weak relatives share a locally isometric Kähler geometry, but not necessarily the same complex structure on the shared piece (Placini, 2023).

The dimension assumption $M_2$0 is essential. In complex dimension $M_2$1, any isometry is automatically holomorphic or anti-holomorphic, so the weak notion does not differ from the classical one. In higher dimension, however, the two notions separate: the paper points out that in real dimension $M_2$2, a Riemannian manifold may carry a continuous family of parallel complex structures, and if two such complex structures lie in different isometry-group orbits, the resulting Kähler manifolds are weak relatives but not relatives. Hyperkähler manifolds provide such examples (Placini, 2023).

The geometric distinction is therefore precise. For relatives, the common subgeometry is both metric and complex-analytic. For weak relatives, what is shared is only the local Riemannian structure of the submanifold, with compatibility with the ambient complex structures imposed only after separate holomorphic isometric immersions into $M_2$3 and $M_2$4 (Placini, 2023).

2. Projective rigidity and collapse to genuine relatives

A central rigidity theorem states that if a projective manifold $M_2$5 and a Kähler manifold $M_2$6 are weak relatives, then they are relatives. In this setting, the projective hypothesis is strong enough to recover the common holomorphic geometry from the weaker metric datum, so “weakly related” collapses to “related” (Placini, 2023).

The key local input is the lemma that an isometry $M_2$7 between irreducible Kähler manifolds is either holomorphic or anti-holomorphic whenever $M_2$8 is not Ricci-flat. The proof of the projective theorem uses the de Rham decomposition

$M_2$9

where $X$0 is the Ricci-flat factor. Hulin’s result excludes a holomorphic isometric immersion of the Ricci-flat factor into a projective manifold. On each remaining irreducible factor $X$1, the induced isometry is holomorphic or anti-holomorphic; anti-holomorphic factors are conjugated, and the resulting pieces assemble into a global holomorphic isometry $X$2 (Placini, 2023).

This rigidity has an immediate exclusionary consequence. A projective Kähler manifold $X$3 and a product $X$4 of a flat Kähler manifold with a homogeneous bounded domain are not weak relatives, because they were already known not to be relatives, and in the projective case weak relativeness would force actual relativeness (Placini, 2023).

3. Strict relatives and strong non-relativity

The same literature introduces strict relatives: two Kähler manifolds $X$5 and $X$6 are strict relatives if they are relatives, but there exists no local holomorphic isometry of one into the other. This separates the existence of a common Kähler submanifold from the much stronger condition that one ambient manifold locally holomorphically isometrically immerses into the other (Placini, 2023).

Several nontrivial examples are constructed.

Pair	Shared submanifold	Obstruction to ambient immersion
$X$7 and $X$8 with flat/Fubini–Study product metric	$X$9	Dimensional and curvature reasons
$\varphi_i:X\to M_i$0 with hyperbolic metric and a bounded symmetric domain of rank $\varphi_i:X\to M_i$1	Totally geodesic $\varphi_i:X\to M_i$2	Dimension and symmetric-space rigidity
$\varphi_i:X\to M_i$3 and $\varphi_i:X\to M_i$4 for $\varphi_i:X\to M_i$5	Totally geodesic $\varphi_i:X\to M_i$6	Suyama’s theorem
$\varphi_i:X\to M_i$7 and $\varphi_i:X\to M_i$8 with metric $\varphi_i:X\to M_i$9, $i=1,2$0	A $i=1,2$1 fiber	Calabi rigidity plus fullness in $i=1,2$2

These examples show that strict relativeness occurs in reducible and irreducible, compact and noncompact, flat and non-flat settings. The phenomenon is therefore not an artifact of obvious embeddings; it isolates ambient manifolds that genuinely share Kähler subgeometry without containing one another holomorphically and isometrically (Placini, 2023).

A complementary strengthening is the notion of a manifold strongly not relative to any projective Kähler manifold: $i=1,2$3 has this property if $i=1,2$4 is not relative to any projective Kähler manifold for every $i=1,2$5. Two criteria are given. First, if $i=1,2$6 is infinite projectively induced for all $i=1,2$7 and $i=1,2$8 is not relative to any $i=1,2$9, then $X$0 is strongly not relative. Second, transversally full immersions into $X$1 yield a similar conclusion. Bergman–Hartogs domains and certain Fock–Bargmann–Hartogs domains are then shown to be strongly not relative to any projective manifold (Zedda, 2016).

4. Ambient weakenings: relative amenability and weak relative Dixmier averaging

Outside Kähler geometry, closely analogous weakening mechanisms occur when a property is made relative to an ambient object. For a locally compact group $X$2 and a closed subgroup $X$3, relative amenability means that for every non-empty convex compact $X$4-space $X$5, the subgroup $X$6 fixes a point in $X$7. This is a priori weaker than amenability of $X$8, because the fixed-point requirement is imposed only on $X$9-spaces restricted to $X_1$0, not on all $X_1$1-spaces (Caprace et al., 2013).

The notion admits several equivalent analytic formulations. For a closed subgroup $X_1$2, relative amenability is equivalent to: a bounded right approximate identity for the ideal $X_1$3; the existence of a left $X_1$4-invariant mean on $X_1$5; and the existence of a $X_1$6-equivariant continuous linear map

$X_1$7

which can be chosen positive and of norm one. The paper also solves Reiter’s problem by proving

$X_1$8

and introduces the class $X_1$9 of groups for which relative amenability implies amenability for all closed subgroups. This class contains all familiar groups listed in the paper and is stable under numerous permanence operations, while relative amenability itself is closed under Chabauty limits (Caprace et al., 2013).

A related operator-algebraic weakening appears in the weak relative Dixmier property for inclusions $X_2$0 of von Neumann algebras equipped with a faithful normal semifinite operator-valued weight $X_2$1. For every positive $X_2$2 with $X_2$3, the $X_2$4-weak closure of the convex hull of the $X_2$5-unitary orbit of $X_2$6 intersects the relative commutant $X_2$7. This extends Marrakchi’s expectation-based result to operator-valued weights and is applied to a tracial-free reformulation of Popa’s intertwining criterion, a type III relative solidity theorem, and a Galois-type correspondence for crossed products by totally disconnected groups (Isono, 25 Aug 2025).

5. Weak relatives in Banach-space approximation and compactness

In Banach operator-ideal theory, the phrase “weaker relatives” is used explicitly for the weak $X_2$8-BAP for $X_2$9 and the local $\ge 2$0-BAP for $\ge 2$1, introduced as weakenings of the $\ge 2$2-bounded approximation property for a Banach operator ideal $\ge 2$3. The implication chain is

$\ge 2$4

The weak version requires finite-rank endomorphisms $\ge 2$5 converging pointwise to $\ge 2$6 with $\ge 2$7, while the local version only asks for pointwise approximation of each $\ge 2$8 by finite-rank operators $\ge 2$9. The local notion is strictly weaker in general, but for injective ideals it coincides with the weak one; for injective closed ideals all three ideal-relative notions coincide. The paper further identifies Saphar’s approximation property of order $\varphi_i:X_i\to M_i$0 with the weak/local $\varphi_i:X_i\to M_i$1-BAP for the ideal $\varphi_i:X_i\to M_i$2 of absolutely $\varphi_i:X_i\to M_i$3-summing operators (Lassalle et al., 2014).

A parallel weakening occurs in the theory of $\varphi_i:X_i\to M_i$4-null sequences. For $\varphi_i:X_i\to M_i$5 and $\varphi_i:X_i\to M_i$6, the $\varphi_i:X_i\to M_i$7-convex hull of $\varphi_i:X_i\to M_i$8 is

$\varphi_i:X_i\to M_i$9

and a set is relatively $M_2$00-compact if it lies in such a hull. A sequence $M_2$01 is $M_2$02-null if for every $M_2$03 there exist $M_2$04 and $M_2$05 such that $M_2$06-conv$M_2$07 for all $M_2$08. The omnibus theorem proves the equivalence

$M_2$09

and extends the same pattern to unconditional and weak variants via the operator ideals $M_2$10 and $M_2$11 (Ain et al., 2014).

6. Weak heirs, coheirs, and broader family terminology

In topological dynamics and model theory, weak heirs and weak coheirs compare Ellis semigroups attached to $M_2$12-algebras $M_2$13 and $M_2$14 with $M_2$15. If $M_2$16 and $M_2$17, then $M_2$18 is a weak heir of $M_2$19 when $M_2$20 for every $M_2$21. It is a weak coheir of $M_2$22 when the $M_2$23-behavior of $M_2$24 over $M_2$25 is controlled by restriction. These notions are exactly the compatibility conditions needed to make restriction respect semigroup multiplication: weak heirs correspond to

$M_2$26

whereas weak coheirs correspond to

$M_2$27

Under the hypothesis that every minimal left ideal in $M_2$28 is a group, the Ellis groups of $M_2$29 are isomorphic to closed subgroups of the Ellis groups of $M_2$30. In the stable case, weak heirs and weak coheirs are identified with nonforking conditions on the translated fragments $M_2$31 and $M_2$32 (Malinowski et al., 2022).

The same body of literature also uses “relatives” in a broader family-classification sense rather than as a formal weakening. The survey on pure braid groups treats $M_2$33, $M_2$34, $M_2$35, and $M_2$36 as relatives of $M_2$37, distinguished by resonance varieties, Chen ranks, Koszulness, and formality (Suciu et al., 2016). In regular-map theory, the six Petrie relatives of a map arise from duality and Petrie-duality (Cutler et al., 2012). Over finite fields, relatives of the Hermitian curve are plane curves

$M_2$38

with point-count congruence $M_2$39 and a classification of those having at least two rational inflexions (Homma et al., 2024). In arithmetic, the even values $M_2$40 are described as relatives of Catalan’s constant, and at least one of $M_2$41 is proved irrational (Zudilin, 2018). In topology, the close relatives $M_2$42 of Hilbertian balls are all shown to be homeomorphic whenever $M_2$43, including the positive part and certain half-ball and band-cut variants (Avilés, 2 Jul 2026).

Across these contexts, the recurrent mathematical pattern is stable: a “weak relative” retains enough of the ambient or structural signature of a stronger notion to permit fixed-point theorems, rigidity upgrades, homeomorphism classifications, or semigroup comparisons, while still being formally less restrictive than the original relation.