arXiv:2107.10087v1 [math.DG] 21 Jul 2021 uv,paa edscimrin suogoei,ttlyublcs umbilic totally pseudo-geodesic, immersion, geodesic planar curve, 53C42. and xrni shape extrinsic xrni hp fsml etcre in curves test simple of shape extrinsic feeygoei of geodesic every of that proves one curves, test e od n phrases. and words Key .Agnrlzto fTerm7 Theorem of generalization References results A main the of 6. Proofs pseudo-geodesics 5. Planar curves 4. Planar 3. Preliminaries results main and 2. Introduction 1. ie nioercimmersion isometric an Given 2020 Date Q aua rbe st ecietegoer of geometry the describe to is problem natural a , uy2,2021. 22, July : in of geodesic every hc ersn aua xrni eeaiaino ohgeodes both of being generalization pseudo : extrinsic planar bun mannian called natural normal curves, a the of represent in family which parallel a on is based vector is curvature characterization mean u tot the the that the of such and direction of vector vect curvature characterization curvature mean mean nonzero novel having normalized those a parallel with provide immersions we isometric bilic First, manifolds. ian Abstract. spaa nteodnr es;being sense; ordinary the in planar is fcdmninoeimrin,w rv h olwn ttmn:a statement: following the immersion prove we Moreov d immersions, uniqueness. in codimension-one and curves of existence these local study We their establish relation. particular, linear a satisfy curvatures mal ahmtc ujc Classification. Subject Mathematics R 3 . facurve a of TE AKOSNADMTE RAFFAELLI MATTEO AND MARKVORSEN STEEN SUBMANIFOLDS UMBILIC TOTALLY ι LNRPED-EDSC AND PSEUDO-GEODESICS PLANAR : esuyttlyublcioercimrin ewe - between immersions isometric umbilic totally study We M ֒ M 1. M → nrdcinadmi results main and Introduction sagoei of geodesic a is spaa.Ti xed elkonrsl bu surfaces about result well-known a extends This planar. is stoi meso,nraie encrauevco,planar vector, curvature mean normalized immersion, Isotropic Q γ planar sttlyublci n nyi h xrni hp of shape extrinsic the if only and if umbilic totally is ι in sttlygoei fadol fteetiscshape extrinsic the if only and if geodesic totally is M hi atndvlpeti h agn space tangent the in development Cartan their , ι stecurve the is Contents : M ֒ M pseudo-geodesics Q rmr:5B5 eodr:5A4 53C40, 53A04, Secondary: 53B25; Primary: → 1 ee n nters fteppr the paper, the of rest the in and Here, . o xml,choosing example, For . Q ewe imninmanifolds Riemannian between ι ◦ γ . hi edscadnor- and geodesic their , ubmanifold. ι ( M i etri the in vector nit r ntecase the in er, ) c n Rie- and ics ti n,in and, etail isometric n ≡ -geodesics, l.Such dle. M lyum- ally ι r i.e., or, goeisas -geodesics hog the through M 14 13 9 7 5 4 1 PLANAR PSEUDO-GEODESICS 2

Another fundamental result of this sort is the well-known theorem of Nomizu and Yano [14], characterizing extrinsic by the property that the extrinsic shape of every in M is a circle in Q. Recall that a circle in a is a curve whose Cartan development in the tangent space is an ordinary circle. Extrinsic spheres are totally umbilic (immersed) submanifolds whose mean curvature vector is parallel with respect to the normal connection. A closely related concept, first studied by Chen in [3], is that of totally umbilic submanifold with normalized parallel mean curvature vector. In this case, only a unit vector field in the direction of the (nonzero) mean curvature vector is required to be parallel. A generalization of Nomizu–Yano’s theorem to this broader class of isometric immersions appeared in [17, 1]. In order to present this generalization, we need some preliminaries. Let γ be a smooth unit-speed curve in M, and let κ be its ′ ′ 1/2 geodesic curvature, i.e., κ = γ′ γ , γ′ γ , where denotes the Levi-Civita connection of M. One says thath∇ γ has∇ orderi two at the∇ point γ(s) if there exists a local field of unit vectors Y along γ such that

′ ′ γ (t)γ t=s = κ(s)Y (s), ∇ ′ ′ Y = κ(s)γ (s). ( γ (t) t=s ∇ −

Theorem 1 ([1]). The following statements are equivalent: (i) ι is totally umbilic and, away from geodesic points (i.e., on the open subset where the second fundamental form is nonzero), has parallel normalized mean curvature vector. (ii) For every orthonormal pair of vectors u, v TpM, there exists a curve γ, defined in a neighborhood of 0, such that ∈ ′ ′ (a) γ(0) = p, γ (0) = u, and γ′(0)γ t=0 = κ(0)v; (b) The extrinsic shape ι γ ∇of γ has| order two at ι(p); (c) κ′(0)/κ(0) =κ ˜′(0)/κ˜(0)◦ , where κ˜ is the geodesic curvature of ι γ. ◦ It is clear that Theorem 1 is conceptually rather different than Nomizu–Yano’s classic result, and more difficult to understand geometrically. It is a purpose of this paper to provide an alternative characterization of the same class of submani- folds. Our description has the advantage of being quite intuitive and geometrically appealing. On the other hand, it is based on a family of test curves whose defi- nition depends on the second fundamental form. These are a natural (extrinsic) extension of both geodesics and Riemannian circles, called pseudo-geodesics. Pseudo-geodesics were introduced in the literature by Wunderlich for surfaces embedded in three-dimensional Euclidean space [19, 20]. A curveγ ¯ = ι γ lying on a surface is a pseudo-geodesic if the acceleration vectorγ ¯′′ makes a◦ constant angle θ with the surface normal. Note that the angle is zero precisely when γ is a geodesic. This definition extends straightforwardly to any Riemannian submanifold; in- deed, it is easy to see that the angle θ is constant if and only if either γ is a geodesic or else the ratio between the normal curvature γ¯′′, N and the geodesic curvature κ is constant (Lemma 13). Hence, for Wunderlich’sh definitioni to make sense in arbitrary dimension and codimension, one just needs to interpret the PLANAR PSEUDO-GEODESICS 3 normal curvature as τ = α(¯γ′, γ¯′),α(¯γ′, γ¯′) 1/2, where α is the second fundamen- tal form. In fact, we shallh define pseudo-geodesicsi in any Riemannian manifold equipped with a field of scalar-valued or vector-valued symmetric bilinear forms (Definition 14). On the other hand, in dimension greater than two, given a point p M, a ∈ tangent vector v TpM, and a constant c = 0, the initial value problem for the pseudo-geodesic equation∈ κ = cτ is underdetermined.6 Thus, in order to have a well-posed problem, we consider planar pseudo-geodesics, i.e., pseudo-geodesics that are of order two at all of their points; see Definition 8 and subsequent re- marks. Equivalently, pseudo-geodesics whose Cartan development in the tangent space of M lies in a plane (Proposition 11). Using them as test curves, we will prove our first main result. Theorem 2. If, for some c R 0 , the extrinsic shape of every planar c- pseudo-geodesic of (M, ι∗α) is∈ planar,\{ } then ι is totally umbilic and, away from geodesic points, has parallel normalized mean curvature vector. Conversely, if ι is totally umbilic, then the extrinsic shape of every planar pseudo-geodesic of (M, ι∗α) is planar. Corollary 3. If ι is totally umbilic, then the extrinsic shape of every geodesic of M is planar. Corollary 4 ([13, Theorem 2]). If ι is a non-totally geodesic extrinsic , then the extrinsic shape of every geodesic of M is a circle. Remark 5. If ι is a hypersurface, then the normalized mean curvature vector is automatically parallel. It is well-known that, if an isometric immersion takes planar curves to planar curves, then it is totally geodesic [18, Theorem 1]. A natural question, then, is whether it is possible for the type of immersion considered in Theorem 2 to pre- serve the planarity of additional curves without necessarily being totally geodesic. Our next result answers this question negatively. Proposition 6. Suppose that ι is totally umbilic with parallel normalized mean curvature vector. If a curve has planar extrinsic shape, then it is a pseudo- geodesic. An additional problem that Theorem 2 leaves open is to characterize the sub- manifolds all of whose geodesics have planar extrinsic shape. The particular case where the ambient manifold is a space form was examined in [5, 10, 16, 2, 6]. Here we shall give a complete solution when the codimension is one. Theorem 7. Suppose that M is a hypersurface of Q. If the extrinsic shape of every geodesic of M is planar, then ι is totally umbilic. In particular, if the extrinsic shape of every geodesic of M is a circle, then ι is a non-totally geodesic extrinsic sphere. The paper is organized as follows. The next section presents some preliminar- ies, mostly for the sake of fixing relevant notation and terminology. In section 3, motivated by the notion of Riemannian circle, we introduce planar Riemannian curves, thus generalizing the standard notion of planarity valid in space forms. PLANAR PSEUDO-GEODESICS 4

In section 4 we then define pseudo-geodesic; in particular, by restricting our at- tention to planar pseudo-geodesics, we establish a local existence and uniqueness result. In section 5 we proceed with the proofs of Theorem 2, Proposition 6, and Theorem 7. Finally, in section 6 we extend Theorem 7 to submanifolds of arbitrary codimension.

2. Preliminaries In this section we recall some basic facts that are used throughout the paper. To begin with, let Q be a Riemannian manifold and M Q an immersed submanifold. Identifying, as customary, the tangent space of⊂ M at p with its image under the differential of the inclusion M ֒ Q, we have the orthogonal decomposition → T Q = T M N M, p p ⊕ p where NpM is the normal space of M at p. Under this identification, every smooth vector field X on M can be considered as a vector field X along M, that is, a smooth section of the ambient tangent bundle over M. Let X(M) and X¯(M) denote the sets of smooth vector fields on and along M, respectively. Let X(M)⊥ be the set of normal vector fields along M. Clearly, X¯(M)= X(M) X(M)⊥. If X X(M) and X¯ X¯(M), then ⊕ ∈ ∈ X¯ = π⊤ X¯ + π⊥ X,¯ ∇X ∇X ∇X where is the Levi-Civita connection of Q, π⊤ and π⊥ are the orthogonal pro- e e e jections∇ onto the tangent and normal bundle of M, and where both X and X¯ are extendede arbitrarily to Q. In particular, if X¯ = Y X(M), then ∈ Y = Y + α(X,Y ). ∇X ∇X Here is the Levi-Civita connection of (M, ι∗g˜) and α the second fundamental form.∇ e ⊥ Similarly, if N X(M) , then, denoting by AN the shape operator of M with respect to N and∈ by ⊥ the normal connection of M, ∇ N = A (X)+ ⊥ N. ∇X N ∇X The normal connection allows us to introduce a natural covariant differentiation ∗ for the second fundamentale form, as follows; see [9, p. 231] for more details. Let∇ F be the smooth vector bundle over M whose fiber at each point is the set of bilinear maps TpM TpM NpM. For any smooth section B of F and any X X(M), let ∗ B be× the smooth→ section of F given by ∈ ∇X ( ∗ B)(Y,Z)= ⊥ (B(Y,Z)) B( Y,Z) B(Y, Z). ∇X ∇X − ∇X − ∇X It is standard to prove that ∗ is a connection in F . We next turn our attention∇ to totally umbilic submanifolds. Given any normal vector η NM, we say that M is umbilic in direction η ∈ if the shape operator Aη is a multiple of the identity. If M is umbilic in every normal direction, then M is said to be a totally umbilic submanifold of Q. PLANAR PSEUDO-GEODESICS 5

One can show that M is totally umbilic if and only if, for every pair of vector fields X,Y X(M), the following relation holds between the second fundamental form and the∈ mean curvature vector H of M: α(X,Y )= X,Y H. h i Recall that the mean curvature vector of M is the normal vector field along M given by H = m−1 tr(α), where m = dim M and tr(α) is the trace of α. Equivalently, in terms of a local orthonormal frame (X1,...,Xm) for T M, m −1 H = m α(Xj,Xj). j=1 X Among totally umbilic submanifolds, extrinsic spheres are particularly impor- tant. A totally umbilic submanifold is called an extrinsic sphere if the mean curvature vector is parallel with respect to the normal connection, that is, if ⊥ X X H = 0 for all X (M). Beware that some authors require H to be nonzero. ∇ Our main interest∈ in this paper lies in the family of totally umbilic submanifolds with parallel normalized mean curvature vector, which naturally generalizes that of non-totally geodesic extrinsic spheres. Suppose that H is always different from zero. Then the unit normal vector field H/ H is well-defined. One says that M has parallel normalized mean curvature vectork |if ⊥ (H/ H ) = 0 for all X X(M). ∇X k | ∈ 3. Planar curves In this section we define planar curves in a Riemannian manifold M (M, , ) and extend several well-known results concerning circles. ≡ h· ·i Definition 8. Let γ : I M be a smooth, unit-speed curve, and denote by T its tangent vector. We say→ that γ is planar if there exists a unit vector field Y along γ and a smooth function κ: I R such that → T = κY, (1) ∇T Y = κT. (∇T − Remarks. – In dimension two every regular curve is planar. – A geodesic is a planar curve with κ = 0. – A planar curve with constant κ> 0 is called a circle [14]. – If M has constant sectional curvature, then a regular curve in M is planar if and only if it lies in some two-dimensional, totally geodesic submanifold of M. This follows from Erbacher’s codimension reduction theorem [4]. – A planar curve has order two at every point. Conversely, if a curve has order two at every point and its curvature is never zero, then it is planar. Hence, planar curves generalize Riemannian circles. Nomizu and Yano proved that circles are precisely those curves in M that satisfy the differential equation 2 T + κ2T =0, ∇T PLANAR PSEUDO-GEODESICS 6

1/2 where κ = T T, T T is the geodesic curvature. In the case of planar curves, the followingh∇ lemma∇ holds.i Lemma 9 ([8, Lemma 2.3]). A curve γ is planar if and only if it satisfies the differential equation (2) κ 2 T + κ3T κ′ T =0. ∇T − ∇T Proof. Notice that the lemma holds when γ is a geodesic. Hence, we may assume that κ(t) > 0 for all t I. If γ is planar, then ∈ 2 T = (κY )= κ Y + T (κ)Y ∇T ∇T ∇T = κ2T + κ′Y − ′ T = κ2T + κ ∇T , − κ which shows that equation (2) holds. Conversely, consider an arbitrary unit-speed curve γ, and define a unit vector field Y along γ by T Y = ∇T . κ It follows that T T = κY and ∇ ′ T 2 T 1 Y = ∇T = ∇T + T. ∇T ∇T κ κ κ ∇T     In particular, if (2) holds, then ′ κ 2 ′ T T κ T 1 Y = κ ∇ − + T ∇T κ κ ∇T   = κT, − as desired.  We now characterize planar curves through the notion of development, in the sense of Cartan. Our result extends Nomizu–Yano’s [14, Proposition 3]. The proof is conceptually the same as the one in [14], but is nevertheless included for the reader’s convenience. Definition 10. Let p = γ(u) be an arbitrary point in the image of γ. The Cartan ∗ development of γ in the tangent space TpM is the unique curve γ : I TpM such that → (i) (γ∗)′(u)= T (u); (ii) for all t I, the vector (γ∗)′(t) is parallel—in the Euclidean sense—to the ∈ t parallel transport τu(T (t)) of T (t) from γ(t) to γ(u) along γ. Proposition 11. A curve γ is planar if and only if its development γ∗ in the tangent space TpM is a regular planar curve in the ordinary Euclidean sense. Proof. Assume that γ is planar so that (1) holds. Let ∗ t ∗ t T (t)= τu T (t) and Y (t)= τu Y (t). PLANAR PSEUDO-GEODESICS 7

Since the map τ t : T M T M is linear and τ t+h = τ t τ t+h, we obtain u γ(u) → γ(t) u u ◦ t ∗ ∗ t t+h ∗ ′ T (t + h) T (t) τ τ T (t + h) T (t) (T ) (t) = lim − = lim u t − h→0 h h→0 h t+h  t τt T (t + h) T (t) = τu lim − h→0 h = τ t T (t). u ∇T Likewise, we have ′ (Y ∗) (t)= τ t Y (t). u ∇T By virtue of these identities, (1) implies ′ ′ (3) (T ∗) = κY ∗ and (Y ∗) = κT ∗. − ∗ Clearly, these equations express the fact that γ is a planar curve in TpM. ∗ Conversely, assume that the development γ of γ in TpM is a regular planar curve. Then there exists a unit vector field Y ∗ along γ∗ and a smooth function κ ∗ ∗ ′ t which, together with T =(γ ) , satisfy (3). Since τu is an isomorphism between Tγ(t)M and Tγ(u)M, we obtain (1) from (3).  We conclude this section by proving a global existence theorem, which extends [14, Theorem 1]. Theorem 12. Suppose that M is complete. For any orthonormal pair of vectors x, y TpM and any smooth function κ: R R, there exists a unique planar curve∈γ : R M satisfying (1) and such that→γ(0) = p, T (0) = x, and Y (0) = y. → Proof. By the standard theory of ordinary differential equations, the problem defined by the system (3) and the initial condition γ∗(0) = p, T ∗(0) = x, Y ∗(0) = y, ∗ has unique global solution γ in TpM. Since M is complete, it follows from [7, ∗ p. 172, Theorem 4.1] that there is a curve in M whose development in TpM is γ . By the proof of Proposition 11, this is precisely the desired curve γ. 

4. Planar pseudo-geodesics In 1950 Wunderlich considered a natural extrinsic generalization of a geodesic of a surface ι: S ֒ R3. Noting that a curve γ is a geodesic of S if and only if the ambient acceleration→ γ ¯′′ = (ι γ)′′ is parallel to the surface unit normal N, he called a curveγ ¯ in ι(S) a pseudo-geodesic◦ if the angle θ betweenγ ¯′′ and N is constant. The next lemma characterizes pseudo-geodesics in terms of curvature. Lemma 13. A curve γ¯ in ι(S) is a pseudo-geodesic if and only if, for some constant c R, the geodesic curvature κ and the normal curvature τ satisfy κ = cτ. ∈ Proof. Since a geodesic satisfies the lemma with c = 0, we assume κ = 0. It follows that 6 γ¯′′, N = γ¯′′ cos(θ) h i k k PLANAR PSEUDO-GEODESICS 8 implies τ 2 cos(θ)2 = , κ2 + τ 2 κ2 1 sin(θ)2 = = . κ2 + τ 2 1+(τ/κ)2 Note that θ is constant if and only if sin(θ)2 is. Hence, if κ = 0, thenγ ¯ is a pseudo-geodesic precisely when the ratio τ/κ is constant. 6  Basing on this result, we extend Wunderlich’s definition as follows. Definition 14. Let M be a Riemannian manifold, let Σ be a smooth vector bundle of rank n over M, and let σ be a smooth field of symmetric bilinear forms TpM TpM Σp on M. A unit-speed curve γ : I M is a (c-)pseudo-geodesic of (M,× σ) if there→ exists a unit vector field Y along→γ and a constant c R such that ∈ cσ(T, T )Y if n = 1; T = ∇T c σ(T, T ) Y if n 2. ( k k ≥ Now, a fundamental property of geodesics is that, for any tangent vector x ∈ TpM, there exists a geodesic, defined for t < ǫ for some ǫ > 0, such that ′ | | γx(0) = p and γx(0) = x. Unfortunately, unless dim M = 2, one cannot expect pseudo-geodesics to enjoy the same property. Thus, in order for the corresponding initial value problem to be well-posed, we consider planar pseudo-geodesics. In that case, we can prove the following proposition. Proposition 15. Let p M. For any orthonormal pair of vectors x, y in T M ∈ p and for any constant c R, there exists a planar pseudo-geodesic γx,y of (M, σ), defined for t < ǫ for some∈ ǫ> 0, such that | | γx,y(0) = p, Tx,y(0) = x, Tx,y Tx,y(0) = cσ(x, x)y, ′ ∇ where Tx,y = γx,y. Proof. We first consider the case where n = 1, i.e., where σ is a symmetric two- tensor field on M. Let γ : I M be an arbitrary unit-speed curve in M. Let Y be an arbitrary unit vector→ field along γ. Then, by (1), it is clear that γ is a planar pseudo- geodesic of (M, σ) if and only if there exists c R such that ∈ T = cσ(T, T )Y, ∇T Y = cσ(T, T )T. ∇T − We shall explore how these equations look in coordinates. Suppose, thus, that γ is contained in the domain of a smooth chart (u1,...,um) for M around p. Expanding T and Y in terms of the coordinate frame (∂1 = 1 m ∂/∂u ,...,∂m = ∂/∂u ), we obtain j T = T ∂j, j Y = Y ∂j. PLANAR PSEUDO-GEODESICS 9

Using [9, Proposition 4.6], we compute T = T˙ k∂ + T iT jΓ k ∂ , ∇T k ij k Y = Y˙ k∂ + T iY jΓ k ∂ , ∇T k ij k k where a dot indicates differentiation with respect to t and Γij is assumed to be evaluated along γ. Likewise, expressing σ in terms of the coordinate coframe (du1,...,dum), we get σ = σ dui duj. ij ⊗ It follows that i j σ(T (t), T (t)) = σij(γ(t))T (t)T (t). Summing up, γ is a planar pseudo-geodesic of (M, σ) if and only if, for some c R and every k =1,...,m, the following two equations hold: ∈ (4) T˙ k = cσ T iT jY k Γ k T iT j, ij − ij (5) Y˙ k = Γ k T iY j cσ T iT jT k. ij − ij Together withu ˙ k = T k, equations (4) and (5) define a system of 3m ordinary k k k m differential equations in the 3m unknown functions (u , T ,Y )k=1, which admits k k k m a unique, maximal local solution for any initial condition (u (0), T (0),Y (0))k=1. We next examine the case where n 2. Let (E1,...,En) be a smooth or- thonormal frame for Σ along γ. Then≥ there are symmetric two-tensors σ1 = σ, N ,...,σn = σ, N on M, and h 1i h ni σ = σ1N + σnN . 1 ··· n It follows that σ(T, T ) = σ(T, T ), σ(T, T ) 1/2 k k h i = σs(T, T )N , σs(T, T )N 1/2 h s si s i j s i j 1/2 = σijT T Ns, σijT T Ns 2 2 1/2 = σ1 T iT j + + σ n T iT j , ij ··· ij   and so equations (4) and (5) become  

2 2 1/2 (6) T˙ k = c σ1 T iT j + + σn T iT j Y k Γ k T iT j, ij ··· ij − ij   2   2 1/2 (7) Y˙ k = Γ k T iY j c σ1 T iT j + + σn T iT j T k. ij − ij ··· ij Again, the standard theory of ordinary differential equations  guarantees local existence and uniqueness for the initial value problem defined by (6) and (7). 

5. Proofs of the main results Here we prove the new results stated in section 1. To begin with, it is useful to establish a lemma. PLANAR PSEUDO-GEODESICS 10

Lemma 16. Suppose that, for each orthonormal pair of tangent vectors x, y in TpM, either α(x, y)=0 or α(x, x)= α(y,y)=0. Then the following conclusions hold: (i) α(x, x)= α(y,y) for any orthonormal x, y in T M. ± p (ii) If α(x, x)=0 for some x in TpM, then α vanishes at p. Proof. We first prove (i). If (x, y) is orthonormal, then so is 2−1/2(x + y, x y). Thus, assuming the hypothesis of the lemma, either α(x + y, x y) = 0 or− else α(x + y, x + y) = α(x y, x y) = 0. It is easy to check, using− bilinearity and symmetry, that the first− condition− implies α(x, x) = α(y,y), whereas the second α(x, x)= α(y,y). − Now we prove (ii). Suppose there is a unit vector x1 in TpM such that α(x1, x1) = 0. Then, if (x1,...,xm) is an orthonormal basis of TpM, it follows from (i) that α(xj, xj) = 0 for all j =1,...,m. In fact, α(xj +xk, xj +xk)=0for −1/2 all j, k =1,...,m, because 2 (xj + xk) and xh are orthonormal when h = j, k. Since 6

α(xj + xk, xj + xk)=2α(xj, xk), we conclude that α(xj, xk) = 0 for all j and k. Hence, by bilinearity, α vanishes at p.  Proof of Theorem 2. By Lemma 9, the extrinsic shape of γ is a planar curve in Q precisely when (8)κ ˜ 2 T +˜κ3T κ˜′ T =0, ∇T − ∇T whereκ ˜ = T, T 1/2, and where we identified T = (ι γ)′ with T . Since h∇T ∇T i e e ◦ T = T + α(T, T ), denoting by τ the length of α(T, T ), it follows that ∇T ∇T κ˜ = √κ2 + τ 2e. e e e Let p M. If all directions are asymptotic at p, then we have a geodesic point. On the other∈ hand, if x is a nonasymptotic vector at p, then, for every curve γ such that γ(0) = p and γ′(0) = x there exists an open interval ( ǫ, ǫ) such that τ(t) = 0 in ( ǫ, ǫ). − Assume6 that− x is not asymptotic. Then, in ( ǫ, ǫ), − 2κκ′ +2ττ ′ κ˜(κκ′ + ττ ′) κ˜′ = = , 2˜κ κ2 + τ 2 so that equation (8) becomes κκ′ + ττ ′ (9) 2 T +˜κ2T T =0. ∇T − κ2 + τ 2 ∇T Moreover, by computinge e T = T + α(T, T ), ∇T ∇T 2 T = 2 T + α(T, T )+ α(T, T ), ∇e T ∇T ∇T ∇T we see that (9) is equivalent to e e κκ′ + ττ ′ 2 T + α(T, T )+ α(T, T )+˜κ2T ( T + α(T, T ))=0. ∇T ∇T ∇T − κ2 + τ 2 ∇T e PLANAR PSEUDO-GEODESICS 11

Decomposing into tangent and normal components, we finally obtain κκ′ + ττ ′ (10) 2 T + A T + κ2 + τ 2 T T =0, ∇T α(T,T ) − κ2 + τ 2 ∇T ′ ′ ⊥ κκ + ττ (11) α(T, T )+ α(T, T ) α(T, T )=0. ∇T ∇T − κ2 + τ 2 In particular, if κ = cτ for some c = 0 and γ is planar, then (10) and (11) simplify to 6 2 Aα(T,T )T + τ T =0, τ ′ (12) α(T, T )+ ⊥α(T, T ) α(T, T )=0. ∇T ∇T − τ Using ( ∗ α)(T, T )= ⊥α(T, T ) 2α( T, T ), ∇T ∇T − ∇T we rewrite (12) as τ ′ 3α(T, T )+( ∗ α)(T, T ) α(T, T )=0. ∇T ∇T − τ At t = 0, since ( T ) = cτ(0)Y (0), the last equation specializes to ∇T t=0 1 τ ′(0) α(T (0),Y (0)) = α(T (0), T (0)) ( ∗ α)(T (0), T (0)) . 3cτ(0) τ(0) − ∇T   This equation implies that, given a unit vector x T M that is not asymptotic, ∈ p the value α(x, y) does not depend on y TpM, so long as x, y = 0. In fact, since α(x, y)= α(x, y), it is clear that∈α(x, y) = 0 for everyh x andi y such that x, y = 0.− − h If,i on the other hand, x is asymptotic, then, for each y in the orthogonal complement of x in TpM, either α(x, y)=0or α(y,y) = 0; indeed, if α(y,y) = 0, then α(y, x) = 0 by the previous argument. 6 Summing up, we have shown that, if (12) holds for every nonasymptotic x ∈ TpM, then, for each orthogonal pair of vectors x, y in TpM, either α(x, y)=0 or α(x, x) = α(y,y) = 0. Hence, by Lemma 16, if there is x TpM such that α(x, x) = 0, then p is a geodesic point. ∈ Assume that there is no such vector. It follows that there exists a neighbour- hood U of p in M whose points are nonasymptotic. Applying the lemma in [14, p. 168], we conclude that M is totally umbilic in U and that the normalized mean curvature vector coincides with α = α(T, T )/τ along ι γ. Equation (12) therefore simplifies to ◦ τ ′ ⊥α(T, T ) α(T, T )=0. ∇T − τ Substituting α(T, T )= τα, this becomes ⊥α =0, ∇T as desired. Conversely, suppose that ι is totally umbilic. Note that, if γ is a pseudo- geodesic, then equation (8) gives an identity whenever γ(t) is a geodesic point. Hence we may assume that the mean curvature vector never vanishes. It follows PLANAR PSEUDO-GEODESICS 12 that α(T, T T ) = 0 and α = α(T, T )/τ coincides with the normalized mean curvature∇ vector along ι γ. Moreover, ◦ π⊤ α = α, T T = α, T = τT. ∇T h∇T i −h ∇T i − A straightforward computation would reveal that equations (10) and (11) are now equivalent to e e e κκ′ + ττ ′ 2 T + κ2T T =0, ∇T − κ2 + τ 2 ∇T κκ′ + ττ ′ (13) τ ⊥α + τ ′α τ α =0. ∇T − κ2 + τ 2 Suppose that γ is a c-pseudo-geodesic. If c = 0, then γ is a geodesic. In that ⊥ case the first equation gives an identity whereas the second reduces to T α = 0. On the other hand, if c = 0, then substituting τ = κ/c in the first and ∇κ = cτ in the second gives 6 κ 2 T + κ3T κ′ T =0, ∇T − ∇T ⊥α =0. ∇T Evidently, these two are fulfilled exactly when γ is planar and the normalized mean curvature vector α is parallel.  Proof of Proposition 6. Assume the hypothesis of the proposition. Then, for ⊥α = 0, equation (13) is equivalent to ∇T κ(τ ′κ τκ′)=0. − Assume that ι γ is planar. Then, on the open subset where κ(t) = 0, ◦ 6 τ ′ =0, κ which implies that γ is a pseudo-geodesic,  as desired.  Proof of Theorem 7. Suppose that M is a hypersurface of Q. Let N be a unit normal vector field along M, and denote by h the quadratic form associated to the scalar second fundamental form α( , ), N . Clearly, if γ is a geodesic, then κ˜ = τ and h · · i T = α(T, T ), ∇T so that equation (8) reads e (14) τ α(T, T )+ τ 3T τ ′α(T, T )=0. ∇T − In particular, if τ is strictly positive, then α(T, T ) = τN, and therefore (14) becomes e ± (15) τ 2A(T )+ τ 3T =0, ± being A the shape operator. Sm−1 Let p be the unit sphere in TpM. Assume that equation (15) holds for all unit-speed geodesics originating from p with nonasymptotic tangent vector. Then every such vector is an eigenvector of A. We first show that, if x, y Sm−1 ∈ p PLANAR PSEUDO-GEODESICS 13 are linearly independent and nonasymptotic, then h(x) = h(y). Indeed, if h(x + y) = 0, then ± 6 h(x + y)(x + y)= A(x + y)= A(x)+ A(y)= h(x)x + h(y)y, which implies h(x)= h(y). On the contrary, if x + y is asymptotic, then

0= A(x + y), x + y =(h(x)+ h(y))(1 + x, y ), h i h i Sm−1 implying h(x)= h(y). We conclude that p can be decomposed as the union of the following subsets:−

x Sm−1 A(x), x =0 , { ∈ p |h i } x Sm−1 A(x), x = h(y) , { ∈ p |h i } x Sm−1 A(x), x = h(y) . { ∈ p |h i − } Sm−1 R It is clear that, since x A(x), x is a continuous function p , only one of these sets can be nonempty.7→ h Thisi proves that M is umbilic at p→. 

6. A generalization of Theorem 7 We finally present a generalization of Theorem 7 to the case where the codi- mension is arbitrary. To this end, let us first recall the notion of (totally) isotropic immersion, as introduced by O’Neill in [15].

Definition 17. Let ι: M ֒ Q be an isometric immersion. We say that ι is → isotropic at p M if α(x, x),α(x, x) = λp for all unit vectors x TpM. In particular, if ι is∈ isotropich at every pointi p M, then ι is called a totally∈ isotropic ∈ immersion. A totally isotropic immersion is constant isotropic if λp is constant on M.

Theorem 18. If the extrinsic shape of every geodesic of M is planar, then ι is totally isotropic. In particular, if the extrinsic shape of every geodesic of M is a circle, then ι is a non-totally geodesic constant isotropic immersion.

Remark 19. The second part of this theorem is not new. In fact, Maeda and Sato showed that ι is a non-totally geodesic constant isotropic immersion exactly when ∗ the extrinsic shape of every geodesic of M is a circle and ( X α)(X,X) = 0 for all X X(M) [12, Proposition 3.1]. More generally, constant∇ isotropic immersions are∈ characterized by the property that the extrinsic shape of every circle in M has constant geodesic curvature [11].

Remark 20. In a space form, if the extrinsic shape of every geodesic of M is planar, then ι is constant isotropic, and thereby any such extrinsic shape is either a geodesic or a circle; see [16].

Proof. By [15, Lemma 1], ι is isotropic at p exactly when α(x, x),α(x, y) = 0 h i for every orthonormal pair of vectors x, y in TpM. Obviously, if α(x, x) = 0, then α(x, x),α(x, y) = 0 for every y, and so we may assume that x is not asymptotic. h i PLANAR PSEUDO-GEODESICS 14

Let γ be the (unit-speed) geodesic originating from p with tangent vector T (0) = x. Let Y be the parallel transport of y along γ. Then α(x, x),α(x, y) = α(T, T ),α(T,Y ) (0) h i h i = α(T, T ), Y (0) h ∇T i = α(T, T ),Y (0). −h∇T e i Here we have used the Gauss formula as well as orthogonality of Y and α(T, T ). e We now show that T α(T, T ),Y = 0 if ι γ is planar. Indeed, since γ is a geodesic, equation (8)h gives∇ i ◦ (16) eτ α(T, T )= τ 3T + τ ′α(T, T ), ∇T − which implies T α(T, T ),Y = 0. This proves the first part of the theorem. For the secondh∇ part, assumee i that ι γ is a circle, so that τ > 0 and τ ′ = 0. ◦ Then (16) impliese α(T, T ), T = α(T, T ), T = α(T, T ),α(T, T ) = τ 2. −h∇T i h ∇T i h i The last equality shows that α(T, T ),α(T, T ) is constant along γ, and from here the statemente follows easily. h e i  References 1. Toshiaki Adachi, Curvature logarithmic derivatives of curves and isometric immersions, Differential geometry and its applications, World Sci. Publ., Hackensack, NJ, 2008, pp. 13– 21. 2. Toshiaki Adachi, Makoto Kimura, and Sadahiro Maeda, Real hypersurfaces some of whose geodesics are plane curves in nonflat complex space forms, Tohoku Math. J. (2) 57 (2005), no. 2, 223–230. 3. Bang-yen Chen, Surfaces with parallel normalized mean curvature vector, Monatsh. Math. 90 (1980), no. 3, 185–194. 4. Joseph Erbacher, Reduction of the codimension of an isometric immersion, J. Differential Geom. 5 (1971), 333–340. 5. Sing Long Hong, Isometric immersions of manifolds with plane geodesics into Euclidean space, J. Differential Geometry 8 (1973), 259–278. 6. Yi Hong and Yong-an Qin, Helical immersions of order 2, Southeast Asian Bull. Math. 32 (2008), no. 4, 661–666. 7. Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry, vol. 1, Wiley-Interscience, New York, 1963. 8. Masanori Kˆozaki and Sadahiro Maeda, A characterization of extrinsic spheres in a Rie- mannian manifold, Tsukuba J. Math. 26 (2002), no. 2, 291–297. 9. John M. Lee, Introduction to Riemannian manifolds, second ed., Graduate Texts in Math- ematics, no. 176, Springer, Cham, Switzerland, 2018. 10. John A. Little, Manifolds with planar geodesics, J. Differential Geometry 11 (1976), no. 2, 265–285. 11. Sadahiro Maeda, A characterization of constant isotropic immersions by circles, Arch. Math. (Basel) 81 (2003), no. 1, 90–95. 12. Sadahiro Maeda and Noriaki Sato, On submanifolds all of whose geodesics are circles in a complex space form, Kodai Math. J. 6 (1983), no. 2, 157–166. 13. Katsumi Nomizu, Generalized central spheres and the notion of spheres in , Tohoku Math. J. (2) 25 (1973), 129–137. 14. Katsumi Nomizu and Kentaro Yano, On circles and spheres in Riemannian geometry, Math. Ann. 210 (1974), 163–170. 15. Barrett O’Neill, Isotropic and K¨ahler immersions, Canadian J. Math. 17 (1965), 907–915. PLANAR PSEUDO-GEODESICS 15

16. Kunio Sakamoto, Planar geodesic immersions, Tohoku Math. J. (2) 29 (1977), no. 1, 25–56. 17. Tadashi Sugiyama and Toshiaki Adachi, Totally umbilic isometric immersions and curves of order two, Monatsh. Math. 150 (2007), no. 1, 73–81. 18. Hiromasa Tanabe, Characterization of totally geodesic submanifolds in terms of Frenet curves, Sci. Math. Jpn. 63 (2006), no. 1, 83–88. 19. W. Wunderlich, Raumkurven, die pseudogeod¨atische Linien zweier Kegel sind, Monatsh. Math. 54 (1950), 55–70. 20. Walter Wunderlich, Raumkurven, die pseudogeod¨atische Linien eines Zylinders und eines Kegels sind, Compositio Math. 8 (1950), 169–184.

Department of Applied Mathematics and Computer Science, Technical Uni- versity of Denmark, Matematiktorvet, Building 303B, 2800 Kongens Lyngby, Denmark Email address: [email protected]

Institute of Discrete Mathematics and Geometry, TU Wien, Wiedner Haupt- straße 8-10/104, 1040 Vienna, Austria Email address: [email protected]