# Soul theorem
The soul theorem is a result in [Riemannian geometry](https://www.edgepedia.org/riemannian-geometry), proved by [Jeff Cheeger](https://www.edgepedia.org/jeff-cheeger) and Detlef Gromoll in 1972, stating that every connected, complete, non-compact [Riemannian manifold](https://www.edgepedia.org/riemannian-manifold) of non-negative sectional curvature is diffeomorphic to the normal bundle of a compact, totally geodesic, totally convex submanifold, called the soul of the manifold.[1](https://www.math.stonybrook.edu/geomfest09/Meyer2009.pdf) The theorem reduces the topology of an open (non-compact) manifold with sec ≥ 0 to the topology of a compact core: all of the topology of M is concentrated in the soul, so M is of finite type.[2](https://math.tongji.edu.cn/info/1385/8752.htm)
| Key fact | Detail |
|---|---|
| Statement | A complete noncompact manifold M with sec ≥ 0 contains a compact totally geodesic submanifold S, and M is diffeomorphic to the normal bundle ν(S) of S.[1](https://www.math.stonybrook.edu/geomfest09/Meyer2009.pdf) |
| Soul properties | A soul is compact, totally geodesic, totally convex, and without boundary; total convexity means every geodesic segment with endpoints in S lies in S, not only minimizing ones.[3](https://www.claymath.org/library/monographs/cmim03c.pdf) • [4](https://link.springer.com/article/10.1365/s13291-020-00225-x) |
| Positive curvature | If sec > 0, any soul is a point, and consequently M is diffeomorphic to Rn.[3](https://www.claymath.org/library/monographs/cmim03c.pdf) |
| Provenance | Gromoll and Meyer, Ann. of Math. 90 (1969), 75–90, for the positive-curvature case; Cheeger and Gromoll, Ann. of Math. (2) 96 (1972), 413–443, for the general theorem.[5](https://annals.math.princeton.edu/1969/90-1/p06) • [6](https://link.springer.com/article/10.1007/s00208-016-1457-3) |
| Soul conjecture | Cheeger and Gromoll conjectured that positive curvature at a single point forces the soul to be a point; Grigori Perelman proved this in 1994 (J. Differential Geom. 40, 209–212).[2](https://math.tongji.edu.cn/info/1385/8752.htm) • [7](https://electronicsandbooks.com/edt/manual/Magazine/J/Journal%20of%20Differential%20Geometry/1994/40-1-209.pdf) |
| Retraction | Sharafutdinov constructed a distance non-increasing retraction of M onto its soul.[8](https://www.researchgate.net/publication/2108713_THE_SMOOTHNESS_OF_RIEMANNIAN_SUBMERSIONS_WITH_NON-NEGATIVE_SECTIONAL_CURVATURE) |
| Non-uniqueness | The soul is not necessarily unique, but any two souls are isometric.[4](https://link.springer.com/article/10.1365/s13291-020-00225-x) |
## Statement and meaning
The theorem applies to a connected, complete, non-compact Riemannian manifold (M, g) whose sectional curvature satisfies sec(M, g) ≥ 0. Under these hypotheses M contains a submanifold S ⊂ M with three properties: S is compact and totally geodesic, S is totally convex, and M is diffeomorphic to the total space of the normal bundle of S in M.[3](https://www.claymath.org/library/monographs/cmim03c.pdf) A subset is totally convex if every geodesic segment with endpoints in it lies in the subset, whether or not the segment minimizes distance; this is stronger than being totally geodesic.[9](https://mat.uab.cat/~porti/book.pdf) A soul automatically has positive codimension in M.[3](https://www.claymath.org/library/monographs/cmim03c.pdf)
The diffeomorphism M ≅ ν(S) is the topological content of the theorem. Since a vector bundle deformation retracts onto its zero section, M deformation retracts onto S, and the topology of the open manifold is carried by the compact soul together with the twisting of its normal bundle.[2](https://math.tongji.edu.cn/info/1385/8752.htm) If M is closed rather than open, the soul turns out to be M itself, so the theorem has content only in the open case.[9](https://mat.uab.cat/~porti/book.pdf)
**Positive curvature.** If (M, g) has positive sectional curvature, any soul for it is a point, and consequently M is diffeomorphic to Rn.[3](https://www.claymath.org/library/monographs/cmim03c.pdf) This part of the result goes back to the 1969 Annals paper of Detlef Gromoll and Wolfgang Meyer, *On complete open manifolds of positive curvature*, Ann. of Math. 90(1), 75–90.[5](https://annals.math.princeton.edu/1969/90-1/p06) Cheeger and Gromoll proved the splitting theorem for [Ricci curvature](https://www.edgepedia.org/ricci-curvature) ≥ 0 in 1971 and the soul theorem in 1972.[10](https://neckpinch.com/guide/comparison-convergence-and-heat/splitting-and-soul-theorems/)
## Examples
The standard examples show how the soul records the flat directions of the metric:[4](https://link.springer.com/article/10.1365/s13291-020-00225-x)
- Any point of flat Rn is a soul.
- On the paraboloid, which has non-negative curvature, only the apex is a soul; for any other point there are geodesic loops leaving the point.
- A soul of the round cylinder is a slice S1, reflecting the product structure S1 × R.
The soul is not necessarily unique, but any two souls are isometric.[4](https://link.springer.com/article/10.1365/s13291-020-00225-x)
## The soul conjecture and Perelman's proof
At the end of their 1972 paper, Cheeger and Gromoll asked whether a complete non-compact manifold with K ≥ 0 that is strictly positive at some point must have a soul that is a point, equivalently whether M must be diffeomorphic to [Euclidean space](https://www.edgepedia.org/euclidean-space).[1](https://www.math.stonybrook.edu/geomfest09/Meyer2009.pdf) This became known as the soul conjecture of Cheeger and Gromoll.[2](https://math.tongji.edu.cn/info/1385/8752.htm) Perelman gave a positive answer in 1994 in the paper *Proof of the soul conjecture of Cheeger and Gromoll*, Journal of Differential Geometry 40, 209–212, a short proof presented in full.[1](https://www.math.stonybrook.edu/geomfest09/Meyer2009.pdf) • [7](https://electronicsandbooks.com/edt/manual/Magazine/J/Journal%20of%20Differential%20Geometry/1994/40-1-209.pdf) In the same work Perelman established essential geometric structures relating M to its soul, including that the Sharafutdinov retraction is a C1 Riemannian submersion.[2](https://math.tongji.edu.cn/info/1385/8752.htm) • [11](https://www.sciencedirect.com/science/article/abs/pii/S000187082200202X)
## The Sharafutdinov retraction
Sharafutdinov found a distance non-increasing strong deformation retraction Ψ from a complete open manifold of non-negative sectional curvature onto its soul.[8](https://www.researchgate.net/publication/2108713_THE_SMOOTHNESS_OF_RIEMANNIAN_SUBMERSIONS_WITH_NON-NEGATIVE_SECTIONAL_CURVATURE) This retraction organizes the topology at infinity as a metric collapse onto the compact core, leaving room for twisted normal-bundle topology but not arbitrary end topology.[12](https://androma.org/page/Geometric%20Analysis%20I%3A%20Comparison%20Geometry) In the Riemannian setting the retraction is smooth near the soul: Perelman showed that the tubular neighborhood of radius r around S is convex for 0 ≤ r ≤ r* below the focal radius of S, and the Sharafutdinov retraction sh: Dr* → S is C∞ there.[6](https://link.springer.com/article/10.1007/s00208-016-1457-3)
## Non-uniqueness of souls across metrics
A fixed manifold can carry different non-negatively curved metrics whose souls are not even homeomorphic. There is a diffeomorphism Σ7 × R8 ≅ S7 × R8, where Σ7 is an exotic sphere; equipping this manifold with product metrics coming from different non-negatively curved metrics on the spheres yields metrics with non-diffeomorphic souls.[4](https://link.springer.com/article/10.1365/s13291-020-00225-x) Belegradek showed that S3 × S4 × R5 possesses infinitely many pairwise non-homeomorphic souls, and the Belegradek–Kwasik–Schultz theorem extends this: for each k ≥ 3, there are infinitely many complete metrics of non-negative sectional curvature on N = S4 × Sk × R4 whose souls are pairwise non-homeomorphic.[4](https://link.springer.com/article/10.1365/s13291-020-00225-x) González and Zibrowius strengthened the picture further by producing simply-connected open manifolds with a pair of non-homeomorphic souls of positive sectional curvature.[4](https://link.springer.com/article/10.1365/s13291-020-00225-x)
## The converse question
The soul theorem says every open non-negatively curved manifold is a vector bundle over its soul; the converse asks which vector bundles over a compact non-negatively curved base admit a complete non-negatively curved metric on their total space.[4](https://link.springer.com/article/10.1365/s13291-020-00225-x) The answer is not always yes: there exist examples of vector bundles over base spaces with infinite fundamental group whose total space admits no complete metric of non-negative curvature, while no obstructions are known in the case of finite fundamental group.[4](https://link.springer.com/article/10.1365/s13291-020-00225-x) On the positive side, all real vector bundles over Sn for 2 ≤ n ≤ 5 admit non-negative curvature, and Rigas proved that for every real vector bundle E → Sn, with n arbitrary, there is some k such that E × Rk admits a metric of non-negative curvature, an existence statement up to stabilization with trivial bundles.[4](https://link.springer.com/article/10.1365/s13291-020-00225-x)
## Extensions to Alexandrov spaces
An Alexandrov space is a complete length space whose curvature bound is defined by triangle comparison with a model surface; complete Riemannian manifolds and their limits are included in this class.[13](https://link.springer.com/article/10.1007/s12220-024-01870-9) Perelman generalized the Cheeger–Gromoll soul construction to Alexandrov spaces, defining a soul as a nonempty, totally convex, compact subset without boundary, and showed that any complete non-compact Alexandrov space of non-negative curvature homotopically retracts to a soul via the Sharafutdinov retraction.[13](https://link.springer.com/article/10.1007/s12220-024-01870-9) • [11](https://www.sciencedirect.com/science/article/abs/pii/S000187082200202X) The full soul theorem, however, does not generalize: Perelman gave an example of a finite-dimensional Alexandrov space of nonnegative curvature without boundary that is not homeomorphic to the total space of a locally trivial fiber bundle over its soul, so unlike the Riemannian case the local topology of the space is not determined by its soul.[13](https://link.springer.com/article/10.1007/s12220-024-01870-9) The Alexandrov soul conjecture, that a complete non-compact Alexandrov space of non-negative curvature with positive curvature around one point has a soul that is a point, has been proved in dimension 4.[14](https://arxiv.org/abs/1802.04992)
## References
1. [Detlef Gromoll and Wolfgang Meyer (1969). On complete open manifolds of positive curvature. Annals of Mathematics 90(1), 75–90.](https://annals.math.princeton.edu/1969/90-1/p06)
2. [A Survey on Cheeger-Gromoll's Soul Theorem and the Most Recent Developments, Tongji University School of Mathematical Sciences (Shi-Cheng Xu, 2021).](https://math.tongji.edu.cn/info/1385/8752.htm)
3. [The soul theorem, Theorem 2.7, Clay Mathematics Institute monograph.](https://www.claymath.org/library/monographs/cmim03c.pdf)
4. [Bounds on Sectional Curvature and Interactions with Topology, Jahresbericht der DMV (Springer).](https://link.springer.com/article/10.1365/s13291-020-00225-x)
5. [Wolfgang Meyer, Stony Brook Geometry Festival notes (2009), Theorem 11.1.](https://www.math.stonybrook.edu/geomfest09/Meyer2009.pdf)
6. [Strongly nonnegative curvature, Mathematische Annalen (Springer).](https://link.springer.com/article/10.1007/s00208-016-1457-3)
7. [G. Perelman (1994). Proof of the soul conjecture of Cheeger and Gromoll. J. Differential Geometry 40, 209–212.](https://electronicsandbooks.com/edt/manual/Magazine/J/Journal%20of%20Differential%20Geometry/1994/40-1-209.pdf)
8. [The smoothness of Riemannian submersions with non-negative sectional curvature (ResearchGate).](https://www.researchgate.net/publication/2108713_THE_SMOOTHNESS_OF_RIEMANNIAN_SUBMERSIONS_WITH_NON-NEGATIVE_SECTIONAL_CURVATURE)
9. [Manifolds with nonnegative curvature, Section B.2 (UAB lecture notes).](https://mat.uab.cat/~porti/book.pdf)
10. [Splitting and Soul Theorems, Neckpinch Guidebook 9B.5.](https://neckpinch.com/guide/comparison-convergence-and-heat/splitting-and-soul-theorems/)
11. [The Soul Conjecture in Alexandrov Geometry in dimension 4, Advances in Mathematics (ScienceDirect).](https://www.sciencedirect.com/science/article/abs/pii/S000187082200202X)
12. [Geometric Analysis I: Comparison Geometry, Sharafutdinov retraction.](https://androma.org/page/Geometric%20Analysis%20I%3A%20Comparison%20Geometry)
13. [The Injectivity Radius of Souls of Alexandrov Spaces, Journal of Geometric Analysis (Springer).](https://link.springer.com/article/10.1007/s12220-024-01870-9)
14. [The Soul Conjecture in Alexandrov Geometry in dimension 4, arXiv:1802.04992.](https://arxiv.org/abs/1802.04992)
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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Geometry and topology › Differential geometry*
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