GromovlawsonEdit
Gromov-Lawson refers to a landmark collaboration in differential geometry and topology, culminating in a surgery-theoretic approach to when a smooth manifold can support a metric of positive scalar curvature. The work is named for Mikhail Gromov and Herbert B. Lawson Jr., two highly influential figures in global differential geometry. Their methods provide constructive tools for building metrics with positive scalar curvature (PSC) and connect geometric properties to the underlying topology of the manifold Mikhail Gromov Herbert B. Lawson Jr. positive scalar curvature surgery.
The core idea is that PSC can be preserved under certain topological operations, a theme that opened up systematic ways to generate PSC metrics on a wide class of manifolds. This fits into a broader program in which people try to understand which manifolds admit PSC metrics and how those metrics behave under cuts and pastes along submanifolds. The results do not merely give existence proofs; they supply explicit geometric constructions that can be adapted and combined with other techniques from spin manifold theory, K-theory and related areas of modern geometry and topology Rosenberg index KO-theory.
Background
Scalar curvature is a scalar-valued invariant of a Riemannian metric that encapsulates, in a single number at each point, how the metric curves space. A manifold that admits a metric with positive scalar curvature is said to have PSC. The question of which manifolds support PSC metrics ties together local geometric structure with global topological constraints. Gromov and Lawson approached this by examining how surgery—a classical topological operation that removes a sphere times a disk and glues in a disk times a sphere—affects the ability to equip the resulting manifold with PSC. The surgery operation in question is performed along an embedded S^k in an n-manifold, with the key constraint being the codimension of the surgery, namely n−k. The Gromov-Lawson framework shows that when this codimension is at least 3 (i.e., k ≤ n−3), PSC can be maintained through the surgery process.
This perspective complements obstruction-based results, which identify when PSC cannot exist due to global topological invariants. The interplay between constructive techniques (to build PSC metrics) and obstructions (to rule out PSC) has made the Gromov-Lawson approach a central pillar in the study of PSC metrics. The techniques also interact with other programs in geometric analysis, including the study of bordisms and index theory, where the notions of concordance and deformation of metrics come into play bordism index theory.
The Gromov-Lawson surgery theorem
The main result can be stated in a compact form as follows. Let M^n be a compact smooth manifold with a Riemannian metric whose scalar curvature is everywhere positive. Suppose S^k ⊂ M^n is an embedded sphere with trivial normal bundle and consider the surgery that replaces S^k × D^{n−k} by D^{k+1} × S^{n−k−1}, producing a new manifold M'^n. If the codimension n−k is at least 3, then M' admits a Riemannian metric of positive scalar curvature. In other words, PSC is preserved under such a surgery.
Key ideas in the construction include the design of a transition region—often described as a “neck”—that smoothly connects the original PSC metric on M outside the surgery neighborhood to a standard PSC metric on the surgery region. A particularly important ingredient is the use of a torpedo-like metric on product regions (the torpedo metric on S^p × D^q serves as a model for the neck) to ensure the scalar curvature remains positive throughout the modification. This careful local-to-global synthesis is what makes the theorem robust and broadly applicable torpedo metric surgery (topology).
The theorem has several important corollaries and correlative results. It implies, for instance, that if a manifold can be built from simpler pieces via a sequence of codimension-at-least-3 surgeries, and if one of the pieces carries a PSC metric, then many of the resulting manifolds will also carry PSC metrics. This constructive pathway has been used to populate large classes of manifolds with PSC metrics and to explore the boundaries of what topology permits or forbids PSC geometry.
Extensions, relations, and ongoing debates
The Gromov-Lawson framework interacts with broader themes in the study of PSC metrics, including spin geometry, index theory, and nontrivial fundamental groups. In particular, the existence of PSC metrics on spin manifolds is closely tied to index-theoretic obstructions, which can be understood through invariants such as the Rosenberg index in KO-theory of group C*-algebras. The tension between the constructive surgery approach and index-theoretic obstructions has driven a substantial line of research, yielding both positive results and nuanced counterexamples in broader settings Rosenberg index KO-theory C*-algebras.
A prominent strand of the conversation is the Gromov-Lawson-Rosenberg program, which posits a precise link between PSC metrics and KO-theory obstructions attached to the fundamental group of the manifold. While the program has generated a large body of work and deep insights, its general validity has been refined over time. In some cases, the predicted equivalence between the vanishing of a KO-theory index and the existence of a PSC metric has been verified; in other settings, counterexamples and refinements have shown that the relationship is more subtle than originally conjectured. The ongoing developments in this area illustrate a productive tension between constructive geometry and algebraic-topological invariants, a dynamic that continues to shape modern research in PSC geometry spin manifold bordism Gromov-Lawson.
The field also engages with questions about the extent to which PSC metrics can be classified up to concordance or isotopy, and how symmetry and group actions affect the landscape of PSC metrics on manifolds. The techniques introduced by Gromov and Lawson have influenced subsequent methods for building and deforming metrics, and they have served as a blueprint for other geometric analysts exploring curvature, topology, and global differential geometry concordance.
Contemporary discussions in this domain sometimes emphasize methodological debates about the relative value of purely geometric constructions versus index-theoretic obstructions. Proponents of the constructive approach argue that explicit geometric models provide tangible intuition and usable tools for constructing metrics on a wide array of manifolds. Critics, noting the reach of abstract invariants in KO-theory and C*-algebras, stress that understanding the full obstruction picture is essential to a complete picture of PSC geometry. The dialogue remains lively as researchers refine conjectures, identify precise hypotheses, and map the boundary between what surgery can achieve and what index-theory phenomena prevent.
Techniques and impact
The Gromov-Lawson approach is characterized by explicit metric constructions rather than purely abstract existence proofs. One builds local models for the metric near the surgery site and stitches them to the global metric in a way that preserves positivity of scalar curvature. The neck construction and the torpedo-metric paradigm have become standard motifs in PSC geometry, guiding not only surgeries but also related problems where one seeks to control curvature while performing topological modifications. The practical payoff is a toolkit for producing PSC metrics on a broad spectrum of manifolds, including many high-dimensional examples that would be inaccessible by more direct geometric arguments.
Beyond pure existence results, the methods influence adjacent fields where curvature plays a role in topological and geometric analysis. The ideas have cross-pollinated with the study of moduli spaces of metrics, the analysis of geometric flows in a broad sense, and the examination of how curvature interacts with symmetry. The Gromov-Lawson framework stands as a benchmark for how topological operations can be reconciled with geometric constraints, a paradigm that continues to shape how mathematicians think about the global consequences of local curvature conditions manifold differential geometry.