{'fraction_non_alphanumeric': 0.043296747693758, 'fraction_numerical': 0.01111036293852266, 'mean_word_length': 3.8807098258297734, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 2, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Characterizations of umbilic points of isometric immersions in Riemannian and Lorentzian manifolds

Several characterizations of umbilic points of submanifolds in arbitrary Riemannian and Lorentzian manifolds are given. As a consequence, we obtain new characterizations of spheres in the Euclidean space and of hyperbolic spaces in the Lorentz-Minkowski space. We also prove the Lorentzian version of a classical result by Cartan.

characterization of this concept by the study of the intersections of the submanifold with normal submanifolds which are totally geodesic at this point, which provides a natural extension of the notion of umbilic point of a regular surface in the three-dimensional Euclidean space. In this direction, when the submanifold has codimension one, we also get a result on the mean curvature of the hypersurface at a given point. As an application, we prove new characterizations of the sphere in the Euclidean space, as well as of the hyperbolic space in the Lorentz-Minkowski space. In particular, we get that the n-sphere with n ≥ 3 is the only hypersurface in the Euclidean space such that its intersections at each point with 2 normal hyperplanes are (n − 1)-spheres. Analogously, we prove that the n-hyperbolic space with n ≥ 3 is the only spacelike hypersurface in the Lorentz-Minkowski space such that its intersections at each point with 2 normal hyperplanes are (n − 1)-hyperbolic spaces. The first result is a wide generalization of Theorem 1 in [7].
We also prove the Lorentzian version of a classical result by Cartan. It asserts that for dimension bigger than 2 and fixed codimension r bigger than 1, the only Lorentzian manifolds for which for each point the existence of a totally geodesic r-submanifold with an arbitrary prescribed spacelike or timelike tangent space is assured, are those of constant sectional curvature.
The paper is organized as follows. In Section 2 we present the basic background and we get the Lorentzian version of Cartan's result. The main results are shown in Section 4, and the computations needed for the proofs can be found in Section 3. In the last section we apply the main results to the Euclidean space and the Lorentz-Minkowski space, in order to get characterizations of the sphere and the hyperbolic plane.

Preliminaries and a first result
Let (M n+m , g) be a Riemannian manifold and consider an isometric immersion x : (Σ m , g) −→ (M , g). We denote by ∇ the Levi-Civita connection of M and by ∇ Σ the induced connection on Σ.
Hence, the Gauss formula for Σ reads as follows, where II Σ denotes the corresponding second fundamental form of Σ and X, Y are vector fields on Σ.
We define the mean curvature vector H(p) of the immersion at p as If Σ is a hypersurface, we can chose (at least locally) a unitary normal vector field to the hypersurface. Then, the second fundamental form is proportional to it, and so does the mean curvature vector field, whose proportionality constant is called the mean curvature of the hypersurface.
The (immersed) submanifold is called totally geodesic at p ∈ Σ if every M , for all t in some neighborhood of 0, [15]. The immersion x is called totally geodesic if it is totally geodesic for all p ∈ Σ. Of course, if x is totally geodesic at p, then its second fundamental form at p vanishes.
Whereas totally geodesic immersions of dimension bigger than 1 and codimension bigger than 0 are the exception, there always exist (local) submanifolds which are totally geodesic at a single point p ∈ M with an arbitrary prescribed tangent space T ⊂ T p M . Indeed, choose r > 0 so small that the exponential map exp p |B(0,r) is injective. Then, exp p (B(0, r) ∩ T ) is such a submanifold. For dimension bigger than 2 and fixed codimension bigger than 1, the only manifolds for which for each point the existence of a totally geodesic submanifold with an arbitrary prescribed tangent space is assured, are those of constant sectional curvature. This result was proved by Cartan in [10]. As well as the Codazzi equation for a non-degenerate submanifold of M , Σ, where R is the curvature tensor of M , X, Y, Z are vector fields on Σ, ∇ ⊥ is the normal connection of Σ and Theorem 1 For dimension bigger than 2 and fixed codimension r bigger than 1, the only Lorentzian manifolds for which for each point the existence of a totally geodesic r-submanifold with an arbitrary prescribed spacelike or timelike tangent space is assured, are those of constant sectional curvature.
The proof follows the ideas in [12].
Proof: Let (M n+1 , g) be a Lorentzian manifold with 2 ≤ n and fixed 2 ≤ r ≤ n in such a way such that for each point the existence of a totally geodesic rsubmanifold with an arbitrary prescribed spacelike or timelike tangent space is assured.
From Schur's lemma, see [19], it is enough to show that at each point the sectional curvature is constant on the non-degenerate planes of the tangent space.
We fix p ∈ M and consider orthonormal x, y, z ∈ T p M . There exists a totally geodesic r-submanifold such that x, y belong to its tangent plane at p and z is orthogonal to it. Since the submanifold is totally geodesic, its second fundamental form vanishes, and the Codazzi equation assures us that g(R(x, y)z, x) = 0.
Let x, y, z ∈ T p M be spacelike orthonormal vectors. Then where K(x, y) is the sectional curvature of the plane spanned by x and y, y ′ = 1 Let x, y, z ∈ T p M be orthonormal vectors, such that x, y are spacelike and z is timelike. Then where y ′ = 1/2 y + 3/2 z and z ′ = 3/2 y + 1/2 z.
The proof ends with the following two statements.
For each two non-degenerate planes, there exists a chain of planes from one to the other, such that in pairs they span a Lorentzian 3-dimensional space.
For each pair of non-degenerate planes in a Lorentzian space of dimension 3, there exists a chain of non-degenerate planes from one to the other, such that in pairs they intersect orthogonally and both planes of each pair are spacelike, or one is spacelike and the other timelike.
To prove the first statement, it is enough to consider the case of a Lorentzian plane, Π 1 , and an spacelike plane, Π 2 . Take {u 1 , u 2 } an orthonormal base of Π 1 such that u 1 is timelike, u 3 a unitary vector belonging to {u 2 } ⊥ ∩ Π 2 and {u 4 , u 5 } an orthonormal base of Π 2 such that g(u 1 , u 4 ) = 0. It is clear that It is easy to check that Π 1 + Π 3 and Π 2 + Π 3 are non-degenerate timelike 3-dimensional spaces.
To prove the second one, for each pair of non-degenerate planes in a 3dimensional space, Π 1 and Π 2 , we consider the plane generated by their orthogonal vectors, Π 1,2 . We need to distinguish cases. If both planes are spacelike,  Let X, Y be two vector fields on S M . Taking into account that M is totally geodesic at q, we have Moreover, the respective orthogonal projections satisfy Hence, by comparing the Gauss formula for Σ and for S M , we deduce the following equality

Main Results
When studying regular surfaces in we assume that for each (s + n)-submanifold M such that it is totally geodesic at q and T ⊥ q Σ ⊂ T q M , we get the same mean curvature vector at q as that of the isometrically immersed s-submanifold S M := M ∩ Σ. Then, the point q ∈ Σ is umbilic.
Proof: For any two arbitrary unitary vectors u, u ′ ∈ T q Σ, take u 2 , ..., u s ∈ T q Σ such that {u, u 2 , ..., u s } and {u ′ , u 2 , ..., u s } are orthonormal sets, and consider the totally umbilic submanifolds at q spanned by them and T ⊥ q Σ, M and M ′ respectively.
Our hypothesis guaranty that and as a direct consequence of (3) we obtain II Σ (u, u) = II Σ (u ′ , u ′ ).
Note that for m = 2 and codimension 1, Theorem 2 holds trivially. In this case, the curves S M must be considered as 1-dimensional submanifolds immersed in Σ.
When we focus on the case of codimension 1, thanks to (1) and (2), by using the Gauss formula for S M in M , we can deduce that Then, the mean curvature vector field at q of the submanifold Σ ∩ M coincides with the mean curvature vector field of Σ at q, for each (s + n)-submanifold M such that it is totally geodesic at q and T ⊥ q Σ ⊂ T q M .
Proof: The proof follows from (3) and the definition of the mean curvature vector.
As well as the converse of Corollary 3. And finally, as a consequence of (3), we get two more results. They are the first one for submanifolds and the second one for hypersurfaces.
where X, Y ∈ X(S M ). If we suppose that the point q ∈ S M is umbilic, then where H(q) denotes the mean curvature vector field of S M at q.
Thanks to (3), H(q) does not depend on M , and so, q is also umbilic in Σ.
In fact, the hypothesis of the previous theorem can be weakened.

Remark 9
Consider an arbitrary base {w 1 , ..., w m } for T q Σ and take the submanifolds {M α } α which are totally geodesic at q and are spanned by T ⊥ q Σ and any subset of {w 1 , ..., w m } with cardinal s ≥ 2. If we suppose that the point q is umbilic in S Mα for all α, then q is also umbilic in Σ.
In the particular case of hypersurfaces cut by hypersurfaces, only two cuts are needed. Observe that the converse of Theorem 8 and that of Theorem 10 hold trivially.

Spacelike hypersurfaces in Lorentzian manifolds
Suppose now that (M n+m , g) is a Lorentzian manifold and Σ m is a spacelike submanifold of M . Let q ∈ Σ be an arbitrary point and let T ⊂ T q Σ be a subspace of T q Σ. Making use of the exponential map, we can find a Lorentzian submanifold M in M through q, with T q Σ = span T ∪ T ⊥ q Σ such that M is totally geodesic at q. Thus, the submanifold S M = M ∩ Σ is spacelike.
In this setting, the set up and all the results of this section remain true.

For submanifolds of the Euclidean space
In the sequel, we say that an affine subspace of the Euclidean space through a point is normal to a submanifold at that point, if it contains the normal subspace to the submanifold at the point.
A first consequence is obtained from Theorem 8 and Remark 9.

Corollary 11
Let Σ be an m-dimensional submanifold of R m+n (m ≥ 3), q ∈ Σ and 2 ≤ s ≤ m − 1 fixed. Consider an arbitrary base of T q Σ and the family consisting of all the (s + n)-dimensional normal affine subspaces through q spanned by the vectors of the base, {Π i } i∈I . If q ∈ Σ ∩ Π i is umbilic in Π i for all i ∈ I, then q is a umbilic point of Σ.
In particular, we can state the following characterization. We deal now with totally umbilic submanifolds. An m-dimensional nontotally geodesic, totally umbilic submanifold of a Euclidean (m + n)-space is contained in a sphere of an affine (m + 1)-subspace, [11]. Hence, from now on we can focus on hypersurfaces.
As a consequence of Remark 9 we can also enunciate a characterization of the sphere in the Euclidean space of dimension at least four.

Corollary 13
The sphere S m is the only connected and complete hypersurface of R m+1 (m ≥ 3) such that, for some 2 ≤ s ≤ m fixed, its intersection at each point q with the (s + 1)-dimensional normal affine subspaces through q spanned by an arbitrary base of the tangent space at q, are pieces of spheres.
In the previous result, the completeness can be omitted if we ask the intersections to be spheres. In the particular case of hypersurfaces, Theorem 10 gives us the next result, in which we have not included the completeness as an hypothesis.

Corollary 14
The only connected hypersurface Σ in R m+1 (m ≥ 3) such that at each point q ∈ Σ its intersection with 2 normal hyperplanes are spheres, is the sphere S m .
The connectedness can be omitted as an hypothesis. Assume Σ is not connected. Then we can apply the previous result to each connected component.
We get that Σ is made up of spheres. Then, there are points of Σ such that for any normal hypersurface at the point, the intersection with Σ is not an sphere, but many. Which is a contradiction.
We get a result that constitutes a wide generalization Theorem 1 of [7].

Corollary 15
The only hypersurface Σ in R m+1 (m ≥ 3), such that at each point q ∈ Σ its intersection with 2 normal hyperplanes are spheres, is the sphere Making use of Alexandrov's theorem [4] and Corollary 5, we can include the case m = 2 in the following result.

Corollary 16
The only compact and connected hypersurface Σ in R m+1 such that its intersection at each point q ∈ Σ with 2 orthonormal normal hyperplanes (i.e., its normal vectors at q are orthonormal) are pieces of spheres with radius r, is a sphere with radius r.

For submanifolds of the Lorentz-Minkowski space
The second part of the section is devoted to the Lorentz-Minkowski space. Taking into account Section 2.1, we obtain results for spacelike submanifolds analogous to those in the Euclidean space.

Corollary 17
Let Σ be an m-dimensional spacelike submanifold of L m+n (m ≥ 3), q ∈ Σ and 2 ≤ s ≤ m − 1 fixed. Consider an arbitrary base of T q Σ and the family consisting of all the (s + n)-dimensional normal affine subspaces through q spanned by the vectors of the base, {Π i } i∈I . If q ∈ Σ ∩ Π i is umbilic in Π i for all i ∈ I, then q is a umbilic point of Σ.
In particular, we can state the following characterization.

Corollary 18
Let Σ be an m-dimensional spacelike submanifold of L m+n (m ≥ 3), q ∈ Σ and 2 ≤ s ≤ m − 1 fixed. Consider an arbitrary base of T q Σ and the family consisting of all the (s + n)-dimensional normal affine subspaces through q spanned by the vectors of the base, {Π i } i∈I . If for each i ∈ I the intersection Σ ∩ Π i is a piece of an s-hyperbolic plane, then q is a umbilic point of Σ.
When studying totally umbilic submanifolds, it is possible to focus on the case of hypersurfaces. The reason is that an m-dimensional spacelike non-totally geodesic, totally umbilic submanifold of the Lorentz-Minkowski (m + n)-space is either contained in a hyperbolic space of an affine timelike (m + 1)-subspace or in a sphere of an affine spacelike (m + 1)-subspace, [2].

Corollary 19
The hyperbolic space H m is the only connected and complete spacelike hypersurface of L m+1 (m ≥ 3) such that, for some 2 ≤ s ≤ m fixed, its intersection at each point q with the (s + 1)-dimensional normal affine subspaces through q spanned by an arbitrary base of the tangent space at q, are pieces of hyperbolic spaces.
Reasoning as in the Euclidean case, we get the following result.
Corollary 20 The only spacelike hypersurface Σ in L m+1 (m ≥ 3) such that its intersection at each point q ∈ Σ with two normal hyperplanes are (m − 1)hyperbolic spaces, is the hyperbolic m-space.
Observe that if we consider the standard realization of the hyperbolic space H n (1) given by the upper component of the hyperboloid of two sheets, as a direct consequence of our result it is clear that the intersection of any vectorial timelike hyperplane with H n (1), must be a (n − 1)-hyperbolic space, with mean and sectional curvature equal to 1, i.e, a copy of H n−1 (1) embedded in L n+1 and totally umbilic.