Papers
Papers
On q-convex hypersurfaces in Riemannian manifolds (with G. Colombo)
Abstract: We prove that any closed, convex hypersurface in an $(n+1)$-dimensional Riemannian manifold with $\lceil \frac{n}{2} \rceil$-positive curvature operator is a rational homology sphere with finite fundamental group. The same conclusion holds for any $\lceil \frac{n}{2} \rceil$-convex hypersurface, provided that the mean curvature satisfies a sharp pinching condition. Both results follow from more general vanishing and estimation theorems for the Betti numbers of closed $q$-convex immersed hypersurfaces in $(n+1)$-dimensional Riemannian manifolds, under a lower bound on the average of the smallest $(n-p)$ eigenvalues of the curvature operator.
Journal: ---
Links: [Journal] [arXiv]
On the first eigenvalue of the Hodge Laplacian of submanifolds
Abstract: We prove that equality in a sharp lower bound for the first $p$-eigenvalue of the Hodge Laplacian on closed submanifolds in space forms can occur only on topological spheres, assuming positivity.
Journal: Pacific J. Math. 342 (2026), no. 2, 381–385.
Partial scalar curvatures and topological obstructions for submanifolds (with K. Polymerakis and Th. Vlachos)
Abstract: We investigate specific intrinsic curvatures $\rho_k$ (where $1\leq k\leq n$) that interpolate between the minimum Ricci curvature $\rho_1$ and the normalized scalar curvature $\rho_n=\rho$ of $n$-dimensional Riemannian manifolds. For $n$-dimensional submanifolds in space forms, these curvatures satisfy an inequality involving the mean curvature $H$ and the normal scalar curvature $\rho^\perp$, which reduces to the well-known DDVV inequality when $k=n$. We derive topological obstructions for compact $n$-dimensional submanifolds based on universal lower bounds of the $L^{n/2}$-norms of certain functions involving $\rho_k,H$ and $\rho^\perp$. These obstructions are expressed in terms of the Betti numbers. Our main result applies for any $1\leq k \leq n-1$, but it generally fails for $k=n$, where the involved norm vanishes precisely for Wintgen ideal submanifolds. We demonstrate this by providing a method of constructing new compact 3-dimensional minimal Wintgen ideal submanifolds in even-dimensional spheres. Specifically, we prove that such submanifolds exist in $\Sf^6$ with arbitrarily large first Betti number.
Journal: Rev. Mat. Iberoam. 42 (2026), no. 1, 279–298.
On Einstein submanifolds of Euclidean space (with M. Dajczer and Th. Vlachos)
Abstract: Let the warped product $M^n =L^m\times_\varphi F^{n-m}, n\geq m+3\geq 8,$ of Riemannian manifolds be an Einstein manifold with Ricci curvature ρ that admits an isometric immersion into Euclidean space with codimension two. Under the assumption that also $L^m$ is Einstein, but not of constant sectional curvature, it is shown that $\rho=0$ and that the submanifold is locally a cylinder with an Euclidean factor of dimension at least $n-m$. Hence also $L^m$ is Ricci flat. If $M^n$ is complete, then the same conclusion holds globally if the assumption on $L^m$ is replaced by the much weaker condition that either its scalar curvature $S_L$ is constant or that $S_L \leq (2m-n)\rho.$
Journal: Tohoku Math. J. (2) 77 (2025), no. 2, 229–237
The codimension of submanifolds with negative extrinsic curvature (with M. Dajczer and Th. Vlachos)
Abstract: We prove that a substantial isometric immersion into a space form $f\colon M^n\to \Q_c^{n+p}$ with negative extrinsic curvature and flat normal bundle whose first normal bundle has the lowest possible rank possesses substantial codimension $p=n-1$. This fact is already known in the rather special case when also $M^n$ has constant sectional curvature.
Journal: Results Math. vol. 78 (2023), no. 2, Paper No. 42.
Homology vanishing theorems for pinched submanifolds (with Th. Vlachos)
Abstract: We investigate the geometry and topology of submanifolds under a sharp pinching condition involving extrinsic invariants like the mean curvature and the length of the second fundamental form. Homology vanishing results are given that strengthen and sharpen previous ones. In addition, an integral bound is provided for the Bochner operator of compact Euclidean submanifolds in terms of the Betti numbers.
Journal: J. Geom. Anal. (2022), vol. 32, Paper No. 294, 33 pages.
A class of Einstein submanifolds of Euclidean space (with M. Dajczer and Th. Vlachos)
Abstract: We give local and global parametric classifications of a class of Einstein submanifolds of Euclidean space. The highlight is for submanifolds of codimension two since in this case our assumptions are only of intrinsic nature.
Journal: J. Geom. Anal. (2022), vol. 32, no. 2, Paper No. 64, 20 pp.
On constant curvature submanifolds of space forms (with M. Dajczer and Th. Vlachos)
Abstract: We prove a converse to well-known results by E. Cartan and J. D. Moore. Let $f\colon M^n_c\to\Q^{n+p}_{\tilde c}$ be an isometric immersion of a Riemannian manifold with constant sectional curvature $c$ into a space form of curvature $\tilde c\neq c$, and free of weak-umbilic points if $c>\tilde{c}$. We show that the substantial codimension of $f$ is $p=n-1$ if, as shown by Cartan and Moore, the first normal bundle possesses the lowest possible rank $n-1$. These submanifolds are of a class that has been extensively studied due to their many properties. For instance, they are holonomic and admit B\"{a}cklund and Ribaucour transformations.
Journal: Differential Geom. Appl. (2021), vol. 75, Paper No. 101718, 5 pp.
Isometric immersions with flat normal bundle between space forms (with M. Dajczer and Th. Vlachos)
Abstract: We investigate the behavior of the second fundamental form of an isometric immersion of a space form with negative curvature into a space form so that the extrinsic curvature is negative. If the immersion has flat normal bundle, we prove that its second fundamental form grows exponentially.
Journal: Arch. Math. (Basel) (2021), vol. 116, no. 5, 577--583.
On complete conformally flat submanifolds with nullity in Euclidean space
Abstract: We investigate conformally flat submanifolds of Euclidean space with positive index of relative nullity. Let $M^n$ be a complete conformally flat manifold and let $f\colon M^n\to \R^m$ be an isometric immersion. We prove the following results: (1) If the index of relative nullity is at least two, then $M^n$ is flat and $f$ is a cylinder over a flat submanifold. (2) If the scalar curvature of $M^n$ is non-negative and the index of relative nullity is positive, then $f$ is a cylinder over a submanifold with constant non-negative sectional curvature. (3) If the scalar curvature of $M^n$ is non-zero and the index of relative nullity is constant and equal to one, then $f$ is a cylinder over a $(n−1)$-dimensional submanifold with non-zero constant sectional curvature.
Journal: Results Math. (2020), vol. 75, no. 3, Paper No. 106, 7 pp.
Classification of conformally flat isoparametric submanifolds of Euclidean space
Abstract: We provide a direct proof of the complete classification of conformally flat isoparametric submanifolds of Euclidean space.
Journal: Differential Geom. Appl. (2020), vol. 69, 101611, 5 pp.
Conformally flat submanifolds with flat normal bundle (with M. Dajczer and Th. Vlachos)
Abstract: We prove that any conformally flat submanifold with flat normal bundle in a conformally flat Riemannian manifold is locally holonomic, that is, admits a principal coordinate system. As one of the consequences of this fact, it is shown that the Ribaucour transformation can be used to construct an associated large family of immersions with induced conformally flat metrics holonomic with respect to the same coordinate system.
Journal: Manuscripta Math. (2020), vol. 163, no. 3-4, pp. 407--426.
Topological obstructions for submanifolds in low codimension (with Th. Vlachos)
Abstract: We prove integral curvature bounds in terms of the Betti numbers for compact submanifolds of the Euclidean space with low codimension. As an application, we obtain topological obstructions for δ-pinched immersions. Furthermore, we obtain intrinsic obstructions for minimal submanifolds in spheres with pinched second fundamental form.
Journal: Geom. Dedicata (2018), vol. 196, no. 1, pp. 11--26.
Einstein submanifolds with flat normal bundle in space forms are holonomic (with M. Dajczer and Th. Vlachos)
Abstract: We prove that Einstein submanifolds with flat normal bundle in space forms are holonomic. This extends the following well-known result: any isometric immersion with flat normal bundle of a Riemannian manifold with constant sectional curvature into a space form is (at least locally) holonomic, provided that the index of relative nullity vanishes. As an application, when assuming that the index of relative nullity of the immersion is a positive constant we conclude that the submanifold has the structure of a generalized cylinder over a submanifold with flat normal bundle.
Journal: Proc. Amer. Math. Soc. (2018), vol. 146, no. 9, pp. 4035--4038.
Einstein submanifolds with parallel mean curvature
Abstract: We provide a classification of Einstein submanifolds in space forms with flat normal bundle and parallel mean curvature. This extends a previous result due to Dajczer and Tojeiro (Tohoku Math J (2) 45: 43--49, 1993) for isometric immersions of Riemannian manifolds with constant sectional curvature.
Journal: Arch. Math. (Basel) (2018), vol. 110, no. 5, pp. 523--531.
Almost conformally flat hypersurfaces (with Th. Vlachos)
Abstract: We prove a universal lower bound for the $L^{n/2}$-norm of the Weyl tensor in terms of the Betti numbers for compact n-dimensional Riemannian manifolds that are conformally immersed as hypersurfaces in the Euclidean space. As a consequence, we determine the homology of almost conformally flat hypersurfaces. Furthermore, we provide a necessary condition for a compact Riemannian manifold to admit an isometric minimal immersion as a hypersurface in the sphere and extend a result due to Shiohama and Xu (J. Geom. Anal. 7 (1997) 377–386) for compact hypersurfaces in any space form.
Journal: Illinois J. Math. (2017), vol. 61, no. 1-2, pp. 37--51.