# Multi-Layer Softmaxing during Training Neural Machine Translation
for Flexible Decoding with Fewer Layers

Raj Dabre Atsushi Fujita
National Institute of Information and Communications Technology
3-5 Hikaridai, Seika-cho, Soraku-gun, Kyoto, 619-0289, Japan
<EMAIL_ADDRESS>
###### Abstract

This paper proposes a novel procedure for training an encoder-decoder based
deep neural network which compresses $N\times M$ models into a single model
enabling us to dynamically choose the number of encoder and decoder layers for
decoding. Usually, the output of the last layer of the $N$-layer encoder is
fed to the $M$-layer decoder, and the output of the last decoder layer is used
to compute softmax loss. Instead, our method computes a single loss consisting
of $N\times M$ losses: the softmax loss for the output of each of the $M$
decoder layers derived using the output of each of the $N$ encoder layers. A
single model trained by our method can be used for decoding with an arbitrary
fewer number of encoder and decoder layers. In practical scenarios, this (a)
enables faster decoding with insignificant losses in translation quality and
(b) alleviates the need to train $N\times M$ models, thereby saving space. We
take a case study of neural machine translation and show the advantage and
give a cost-benefit analysis of our approach.

##  1 Introduction

Deep neural networks, which allow for end-to-end training, typically consist
of an encoder and a decoder coupled via an attention mechanism. Whereas the
very first deep models used stacked recurrent neural networks (RNN) Sutskever
et al. (2014); Cho et al. (2014); Bahdanau et al. (2015) in the encoder and
decoder, the recent Transformer model Vaswani et al. (2017) constitutes the
current state-of-the-art approach, owing to its better context generation
mechanism via multi-head self- and cross-attentions.

Given an encoder-decoder architecture and its hyper-parameters, such as the
number of layers of encoder and decoder and the sizes of vocabularies (in the
case of text based models) and hidden layers, the parameters of the model,
i.e., matrices and biases for non-linear transformations, are optimized by
iteratively updating them so that the loss for the training data is minimized.
The hyper-parameters can also be tuned, for instance, through maximizing the
automatic evaluation score on the development data. However, in general, it is
not guaranteed (and also highly impossible) that a single set of hyper-
parameters suffices diverse cost-benefit demands at the same time. For
instance, in practical low-latency scenarios, it is often acceptable to
sacrifice output quality for speed. Once a model has been trained, using fewer
number of layers for faster decoding is theoretically possible. Note also that
an optimal set of hyper-parameters does not guarantee that it always results
in the best translation for any input. Hosting multiple models simultaneously
for flexible decoding is impractical, since it requires unreasonably large
quantity of memory.

(a) Multiple tied-layer vanilla models.

(b) Collapsing tied layers into one.

Figure 1: The general concept of multi-layer softmaxing for training multi-
layer neural models with an example of a 4-layer model. Figure 1(a) is a
depiction of our idea in the form of multiple vanilla models whose layers are
tied together. Figure 1(b) shows the result of collapsing all tied layers into
a single layer. The red lines indicate the flow of gradients and hence the
lowest layer in the stack receives the most updates.

To this end, we propose to train multi-layer neural models referring to the
output of all layers during training. Conceptually, this approach equals to
tying the parameters of multiple models with different number of layers, as
illustrated in Figure 1, and is not specific to any type of multi-layer neural
models. In this paper, however, we specifically focus on encoder-decoder
models with $N$ encoder and $M$ decoder layers, and compress $N\times M$
models111Rather than casting the encoder-decoder model into a single column
model with ($N+M$) layers. to update the model, where a total of $N\times M$
losses are computed by softmaxing the output of each of the $M$ decoder
layers, where it attends to the output of each of the $N$ encoder layers. Each
decoder layer is updated referring to a direct signal from the overall loss,
and so does each encoder layer from all the $M$ decoder layers. The number of
parameters of the resultant encoder-decoder model is equivalent to that of the
most complex subsumed model with $N$ encoder and $M$ decoder layers. Yet, we
can now perform faster decoding using a fewer number of encoder and decoder
layers, given that shallower layers are better trained.

In this paper, we take the case study of neural machine translation (NMT) Cho
et al. (2014); Bahdanau et al. (2015), where we focus on the numbers of
encoder and decoder layers of the Transformer model Vaswani et al. (2017), and
demonstrate that it is possible to train a single model with $N$ encoder and
$M$ decoder layers that can be used for decoding with flexibly fewer number of
layers than $N$ and $M$ without appreciable quality loss. We evaluate our
proposed approach on WMT18 English-to-German translation task, and give a
cost-benefit analysis for translation quality vs. decoding speed.

Although we apply our method to encoder-decoder models and evaluate it on an
NMT task, the method should potentially be applicable to any general multi-
layer neural models.

##  2 Related Work

There are studies that exploit multiple layers simultaneously. Wang et al.
(2018) fused hidden representations of multiple layers in order to improve the
translation quality. Belinkov et al. (2017) and Dou et al. (2018) focused on
identifying which encoder or decoder layer can generate useful representations
for different natural language processing tasks. There are also notable
approaches for speeding-up: knowledge distillation Hinton et al. (2015);
Freitag et al. (2017), average attention networks Xiong et al. (2018), and
binary code prediction Oda et al. (2017).

However, to the best of our knowledge, none of them has tackled the issue in
training a flexible translation model.

##  3 Multi-Layer Softmaxing

Figure 1 gives a simple overview of the concept of multi-layer softmaxing for
training a generic 4-layer model. This model takes an input, passes it through
4 layers,222We make no assumptions about the nature of the layers. and then
into the softmax layer to predict the output. Typically, one would apply
softmax to the 4th layer only, compute loss, and then back-propagate gradients
for updating weights. Instead, we propose to apply softmax to each layer,
aggregate the computed losses, and then back-propagate losses. This ensures
that during decoding we can choose any layer instead of only the topmost
layer.

Extending this to a multi-layer encoder-decoder model is straightforward. In
encoder-decoder models, the encoder comprises an embedding layer for the input
(source language for NMT) and $N$ stacked transformation layers. The decoder
consists of an embedding layer and a softmax layer for generating the output
(target language for NMT) along with $M$ stacked transformation layers. Let
$X$ be the input to the $N$-layer encoder, $Y$ the anticipated output of the
$M$-layer decoder as well as the input to the decoder (for training), and
$\hat{Y}$ the predicted output by the decoder. The pseudo-code for our
proposed approach is shown in Algorithm 1. The line 3 represents the process
done by the $i$-th encoder layer, $L^{enc}_{i}$, and the line 5 does the same
for the $j$-th decoder layer, $L^{dec}_{j}$. In simple words, we compute a
loss using the output of each of the $M$ decoder layers which in turn is
computed using the output of each of the $N$ encoder layers. In line 10, the
$N\times M$ losses are aggregated333We averaged multiple losses in our
experiment, but there are a number of options, such as weighted averaging.
before back-propagation. Henceforth, we will refer to this as the $N\times M$
model.

For a comparison, the vanilla model is formulated in Algorithm 2.

Algorithm 1 Training an $N\times M$ model

1:$enc_{0}=X$

2:for $i$ in 1 to $N$ do

3: $enc_{i}=L^{enc}_{i}(enc_{i-1})$

4: for $j$ in 1 to $M$ do

5: $dec_{j}=L^{dec}_{j}(dec_{j-1},enc_{i})$

6: $\hat{Y}=softmax(dec_{j})$

7: $loss_{i,j}=cross\\_entropy(\hat{Y},Y)$

8: end for

9:end for

10:$overall\\_loss=aggregate(loss_{1,1},\ldots,loss_{N,M})$

11:Back-propagate using $overall\\_loss$

$n{\backslash}m$ | 36 individual vanilla models | Our single $N\times M$ model
---|---|---
1 | 2 | 3 | 4 | 5 | 6 | 1 | 2 | 3 | 4 | 5 | 6
1 | 27.07 | 30.25 | 31.63 | 31.61 | 31.48 | 32.11 | 24.24 | 28.85 | 30.24 | 30.55 | 30.91 | 30.93
2 | 29.12 | 32.05 | 32.74 | 32.94 | 33.09 | 32.81 | 27.11 | 31.63 | 33.00 | 33.36 | 33.61 | 33.76
3 | 29.64 | 32.64 | 33.36 | 33.92 | 34.09 | 33.80 | 28.31 | 32.79 | 34.11 | 34.52 | 34.73 | 34.64
4 | 30.29 | 33.61 | 34.33 | 34.44 | 34.16 | 34.39 | 28.96 | 33.30 | 34.48 | 34.76 | 34.81 | 34.76
5 | 31.00 | 33.94 | 34.37 | 35.27 | 34.08 | 34.94 | 29.19 | 33.47 | 34.52 | 34.71 | 34.95 | 34.89
6 | 31.48 | 34.07 | 34.31 | 35.35 | 34.71 | 34.87 | 29.35 | 33.61 | 34.52 | 34.61 | 34.91 | 34.87

Table 1: BLEU scores for the WMT En$\rightarrow$De task. The scores on the
left side are for the 36 individual models that are trained separately. The
scores on the right are for our proposed $N\times M$ model.

$n{\backslash}m$ | 1 | 2 | 3 | 4 | 5 | 6
---|---|---|---|---|---|---
1 | 95.86 | 110.85 | 148.61 | 181.99 | 214.49 | 247.78
2 | 95.47 | 114.44 | 155.78 | 182.00 | 223.05 | 257.47
3 | 92.31 | 114.82 | 153.35 | 192.75 | 225.72 | 265.33
4 | 94.22 | 116.28 | 151.79 | 198.58 | 223.57 | 264.81
5 | 95.38 | 116.57 | 157.17 | 198.48 | 245.10 | 259.94
6 | 94.30 | 117.05 | 155.39 | 195.85 | 241.72 | 264.76

Table 2: Decoding time (in seconds) with the different layer configurations.
Given that a vanilla model with $n$ encoder and $m$ decoder layers and our
$N\times M$ model used with $n$ encoder and $m$ decoder layers have no
difference in the amount of computation, we show only one set of decoding
times.

##  4 Experiments

We trained following two types of models, and evaluated them on both
translation quality and decoding speed.

Vanilla model:

36 vanilla models with 1 to 6 encoder and 1 to 6 decoder layers, trained
referring only to the last layer for computing loss.

$N\times M$ model:

A single $N\times M$ model with $N=6$ encoder and $M=6$ decoder layers,
trained by our multi-layer softmaxing.

###  4.1 Datasets and Preprocessing

We experimented with the WMT18 English-to-German (En$\rightarrow$De)
translation task. We used all the parallel corpora available for WMT18, except
ParaCrawl corpus,444http://www.statmt.org/wmt18/translation-task.html
consisting of 5.58M sentence pairs as the training data and 2,998 sentences in
newstest2018 as test data.

The English and German sentences were pre-processed using the tokenizer.perl
and lowercase.perl scripts in Moses.555http://www.statmt.org/moses

Algorithm 2 Training a vanilla model

1:$enc_{0}=X$

2:for $i$ in 1 to $N$ do

3: $enc_{i}=L^{enc}_{i}(enc_{i-1})$

4:end for

5:for $j$ in 1 to $M$ do

6: $dec_{j}=L^{dec}_{j}(dec_{j-1},enc_{N})$

7:end for

8:$\hat{Y}=softmax(dec_{M})$

9:$loss=cross\\_entropy(\hat{Y},Y)$

10:Back-propagate using $loss$

###  4.2 Model Training

Our multi-layer softmaxing method was implemented on top of an open-source
toolkit of the Transformer model Vaswani et al. (2017) in the version 1.6
branch of tensor2tensor.666https://github.com/tensorflow/tensor2tensor For
training, we used the default model settings corresponding to
transformer_base_single_gpu in the implementation, except what follows. We
used a shared sub-word vocabulary of 32k777We determined the sub-word
vocabularies using the internal sub-word segmenter of tensor2tensor, for
simplicity. and trained the models for 300k iterations. We trained the vanilla
models on 1 GPU and our $N\times M$ model on 2 GPUs with the halved batch size
to ensure that both models see the same amount of training data.

We averaged the last 10 checkpoints saved every after 1k updates, and decoded
the test sentences, fixing a beam size of 4 and length penalty, $\alpha$, of
0.6.888One can realize faster decoding by narrowing down the beam width. This
approach is orthogonal to ours and in this paper we do not insist which is
superior to the other. We evaluated our models using the BLEU metric Papineni
et al. (2002) implemented in tensor2tensor as t2t_bleu: case-sensitive and
detokenized BLEU. We also report on the time (in seconds) consumed to
translate the test set, which includes times for the model creation, loading
the checkpoints, sub-word splitting and indexing, decoding, and sub-word de-
indexing and merging, whereas times for detokenization are not taken into
account.

Note that we did not use any development data for two reasons. First, we train
all models for the same number of iterations.999In our opinion, this is a fair
training method because it ensures that each model sees roughly the same
number of training examples. Second, we use checkpoint averaging before
decoding, where using a development set for early stopping is not needed. We
use this training and decoding approach, because it is known to give the best
results for NMT using the Transformer implementation we use Vaswani et al.
(2017).

###  4.3 Results

Table 2 gives the BLEU scores and Table 2 gives the decoding times of the
models. These summarize the cost-benefit property of our $N\times M$ model in
comparison with the results of the corresponding vanilla models. When our
$N\times M$ model was used for decoding with the 5 encoder and 5 decoder
layers, it achieved a BLEU score of 34.95 which is comparable with the BLEU
score of 35.35 of the best vanilla model with 6-layer encoder and 4-layer
decoder, even though the objective function for our proposed model is
substantially more complex than the one for the vanilla model. Note that the
vanilla models give significantly better results compared to our $N\times M$
models, when using a single decoder layer. However, when the number of decoder
layers are increased there is no statistically significant difference between
the performance of vanilla models and our $N\times M$ model; difference is
less than 1.0 BLEU points in most configurations. We have essentially
compressed 36 models into one.

Regarding the cost-benefit property of our $N\times M$ model, two points must
be noted:

  * •

BLEU score and decoding time increase only slightly, when we use more encoder
layers.

  * •

The bulk of the decoding time is consumed by the decoder, since it works in an
auto-regressive manner. We can substantially cut down decoding time by using
fewer decoder layers which does lead to sub-optimal translation quality.

Consider our $N\times M$ model used with 4 encoder and 3 decoder layers which
gives a BLEU of 34.48. Compared to the best vanilla model (with 6 encoder and
4 decoder layers; 35.35 BLEU), it can decode 1.3 times faster (151.79s vs.
195.85s) for the loss of 0.87 BLEU points. This loss in BLEU is statistically
significant but in real-time low-latency scenarios, however, this certainly
will not have a negative impact on the quality of service.101010There are
several researchers (Tan et al., 2015; Nakazawa et al., 2018) who have shown
that BLEU score is not often correlated with actual translation quality judged
through human evaluation. For instance, when choosing our $N\times M$ model
used with 6 encoder and 2 decoder layers, we lose 1.74 BLEU points, but this
might not have a massive impact on human evaluation. As such, we can choose
this configuration and afford to decode almost twice as fast (117.05s vs
195.85s).

(a) 36 individual vanilla models.

(b) Our single $N\times M$ model.

Figure 2: Distribution of oracle translations among 36 combinations of encoder
and decoder layers for WMT En$\rightarrow$De (2,998 sentences).

One may argue that training a single vanilla model with optimal numbers of
encoder and decoder layers is enough. However, as discussed in Section 1, it
is impossible to know which configuration is the best a priori. More
importantly, a single vanilla model cannot suffice diverse cost-benefit
demands and cannot guarantee best translation for any input (see Section 5.3).
Recall that we aim at a flexible model and all the results in Table 2 have
been obtained using our single $N\times M$ model, albeit using different
number of encoder and decoder layers for decoding.

##  5 Analysis and Discussion

In this section, to better understand the nature of our proposed method, we
give an analysis of our model from the perspective of training times, model
sizes, and decoding behavior, in comparison with vanilla models.

###  5.1 Training Time

All our models were trained for 300k iterations. We thus compare training
times between two models by comparing the time in seconds required to complete
100 iterations of training computations.111111This is the time reported by
tensor2tensor by default. As such, the training time for our $N\times M$ model
was about 9.5 times that of the vanilla model with 6 encoder and 6 decoder
layers. In contrast, the total training time for all individual subsumed 36
vanilla models was 25.54 times121212We measured the collapsed time for a fair
comparison; we assumed that all individual models are trained on a single GPU
one after another, even though one may be able to use 36 GPUs to train the 36
subsumed models in parallel. that of the vanilla model with 6 encoder and 6
decoder layers. Note that this time is calculated by adding the times required
to complete 100 iterations of training computations for all individual vanilla
models. Consequently, our proposed method of training an $N\times M$ model is
computationally much more efficient than independently training all the 36
subsumed models with different number of layers.

###  5.2 Model Size

Our proposed method can help train $N\times M$ models whose number of
parameters is exactly same as vanilla model with $N$ encoder and $M$ decoder
layers. If we train a set of separate models with different numbers of encoder
and decoder layers, we end up with significantly more parameters. For
instance, in case of $N=M=6$ in our experiment, we have 25.16 times more
parameters; a total of 4,600M for 36 subsumed models against 183M for our
$N\times M$ model.

###  5.3 Decoding Behavior

Figure 2 gives the distribution of the test sentences which were best
translated (oracle translations) by different combinations of encoder and
decoder layers during decoding with the vanilla and our $N\times M$ models. We
observed the followings.

  * •

Using the $N\times M$ model, around 50% of the test set is best translated by
using 1 to 2 encoder and decoder layers despite the low corpus-level BLEU
scores with these configurations.

  * •

In contrast, among the individual 36 models, those with 1 to 2 encoder and 1
to 2 decoder layers give the best translation for only 30% of the test set. To
cover 50% of the test set, we have to consider the models up to 3 encoder and
3 decoder layers.

  * •

The distribution of best performing combinations is quite sharp for the
$N\times M$ model unlike the individual vanilla models.

Currently, we do not have an explanation for the difference in behavior
between our and vanilla models in terms of the distribution of optimal layer
combinations. However, it is clear that our $N\times M$ model can essentially
do what 36 individually trained models can do.

In addition, if we can predict an appropriate layer combination to decode each
given input, we can automatically decode with a variable number of layers and
save significant amount of computation. We leave further analyses and the
design of the layer choosing mechanism for future work.

##  6 Conclusion

In this paper, we have proposed a novel procedure for training encoder-decoder
models, where we softmax the output of each of the $M$ decoder layers derived
using the output of each of the $N$ encoder layers. This compresses $N\times
M$ models into a single model that can be used for decoding with a variable
number of encoder ($\leq N$) and decoder ($\leq M$) layers. This model can be
used in different latency scenarios and hence is highly versatile. We have
experimented with NMT as a case study of encoder-decoder models and given a
cost-benefit analysis of our method.

In our future work, we will make an in-depth analysis on the nature of our
$N\times M$ models, such as the diversity of hypotheses generated by different
layers. We will focus on approaches to automatically choose layer combinations
depending on the input and thereby save decoding time by performing the
minimal number of computations to obtain the best output. For further speed up
in decoding as well as model compaction, we plan to combine our approach with
other techniques, such as those mentioned in Section 2. Although we have only
tested our idea for NMT, it should be applicable to other tasks based on deep
neural networks.

## References

  * Bahdanau et al. (2015) Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio. 2015.  Neural machine translation by jointly learning to align and translate.  In _Proceedings of the 3rd International Conference on Learning Representations_ , San Diego, USA.
  * Belinkov et al. (2017) Yonatan Belinkov, Lluís Màrquez, Hassan Sajjad, Nadir Durrani, Fahim Dalvi, and James Glass. 2017.  Evaluating layers of representation in neural machine translation on part-of-speech and semantic tagging tasks.  In _Proceedings of the Eighth International Joint Conference on Natural Language Processing (Volume 1: Long Papers)_ , pages 1–10, Taipei, Taiwan.
  * Cho et al. (2014) Kyunghyun Cho, Bart van Merriënboer, Çaglar Gülçehre, Dzmitry Bahdanau, Fethi Bougares, Holger Schwenk, and Yoshua Bengio. 2014.  Learning phrase representations using RNN encoder–decoder for statistical machine translation.  In _Proceedings of the 2014 Conference on Empirical Methods in Natural Language Processing_ , pages 1724–1734, Doha, Qatar.
  * Dou et al. (2018) Zi-Yi Dou, Zhaopeng Tu, Xing Wang, Shuming Shi, and Tong Zhang. 2018.  Exploiting deep representations for neural machine translation.  In _Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing_ , pages 4253–4262, Brussels, Belgium.
  * Freitag et al. (2017) Markus Freitag, Yaser Al-Onaizan, and Baskaran Sankaran. 2017.  Ensemble distillation for neural machine translation.  _CoRR_ , abs/1702.01802.
  * Hinton et al. (2015) Geoffrey Hinton, Oriol Vinyals, and Jeffrey Dean. 2015.  Distilling the knowledge in a neural network.  _CoRR_ , abs/1503.02531.
  * Nakazawa et al. (2018) Toshiaki Nakazawa, Shohei Higashiyama, Chenchen Ding, Raj Dabre, Anoop Kunchukuttan, Win Pa Pa, Isao Goto, Hideya Mino, Katsuhito Sudoh, and Sadao Kurohashi. 2018.  Overview of the 5th workshop on Asian translation.  In _Proceedings of the 5th Workshop on Asian Translation_ , Hong Kong, China.
  * Oda et al. (2017) Yusuke Oda, Philip Arthur, Graham Neubig, Koichiro Yoshino, and Satoshi Nakamura. 2017.  Neural machine translation via binary code prediction.  In _Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)_ , pages 850–860, Vancouver, Canada.
  * Papineni et al. (2002) Kishore Papineni, Salim Roukos, Todd Ward, and Wei-Jing Zhu. 2002.  BLEU: A method for automatic evaluation of machine translation.  In _Proceedings of the 40th Annual Meeting on Association for Computational Linguistics_ , pages 311–318, Philadelphia, USA.
  * Sutskever et al. (2014) Ilya Sutskever, Oriol Vinyals, and Quoc V. Le. 2014.  Sequence to sequence learning with neural networks.  In _Proceedings of the 27th Neural Information Processing Systems Conference_ , pages 3104–3112, Montréal, Canada.
  * Tan et al. (2015) Liling Tan, Jon Dehdari, and Josef van Genabith. 2015.  An awkward disparity between BLEU / RIBES scores and human judgements in machine translation.  In _Proceedings of the 2nd Workshop on Asian Translation_ , pages 74–81, Kyoto, Japan.
  * Vaswani et al. (2017) Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Łukasz Kaiser, and Illia Polosukhin. 2017.  Attention is all you need.  In _Proceedings of the 30th Neural Information Processing Systems Conference_ , pages 5998–6008, Long Beach, USA.
  * Wang et al. (2018) Qiang Wang, Fuxue Li, Tong Xiao, Yanyang Li, Yinqiao Li, and Jingbo Zhu. 2018.  Multi-layer representation fusion for neural machine translation.  In _Proceedings of the 27th International Conference on Computational Linguistics_ , pages 3015–3026, Santa Fe, USA.
  * Xiong et al. (2018) Deyi Xiong, Biao Zhang, and Jinsong Su. 2018.  Accelerating neural transformer via an average attention network.  In _Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics, Long Papers_ , pages 1789–1798, Melbourne, Australia.


# Planar pseudo-geodesics and
totally umbilic submanifolds

Steen Markvorsen  Department of Applied Mathematics and Computer Science
Technical University of Denmark
Matematiktorvet, Building 303B
2800 Kongens Lyngby
Denmark<EMAIL_ADDRESS>and  Matteo Raffaelli  Institute of Discrete Mathematics
and Geometry
TU Wien
Wiedner Hauptstraße 8-10/104
1040 Vienna
Austria<EMAIL_ADDRESS>
###### Abstract.

We study totally umbilic isometric immersions between Riemannian manifolds.
First, we provide a novel characterization of the totally umbilic isometric
immersions with parallel normalized mean curvature vector, i.e., those having
nonzero mean curvature vector and such that the unit vector in the direction
of the mean curvature vector is parallel in the normal bundle. Such
characterization is based on a family of curves, called planar pseudo-
geodesics, which represent a natural extrinsic generalization of both
geodesics and Riemannian circles: being _planar_ , their Cartan development in
the tangent space is planar in the ordinary sense; being _pseudo-geodesics_ ,
their geodesic and normal curvatures satisfy a linear relation. We study these
curves in detail and, in particular, establish their local existence and
uniqueness. Moreover, in the case of codimension-one immersions, we prove the
following statement: an isometric immersion $\iota\colon M\hookrightarrow Q$
is totally umbilic if and only if the extrinsic shape of every geodesic of $M$
is planar. This extends a well-known result about surfaces in
$\mathbb{R}^{3}$.

###### Key words and phrases:

Isotropic immersion, normalized mean curvature vector, planar curve, planar
geodesic immersion, pseudo-geodesic, totally umbilic submanifold

###### 2020 Mathematics Subject Classification:

Primary: 53B25; Secondary: 53A04, 53C40, 53C42

###### Contents

  1. 1 Introduction and main results
  2. 2 Preliminaries
  3. 3 Planar curves
  4. 4 Planar pseudo-geodesics
  5. 5 Proofs of the main results
  6. 6 A generalization of Theorem 7

##  1\. Introduction and main results

Given an isometric immersion $\iota\colon M\hookrightarrow Q$ between
Riemannian manifolds $M$ and $Q$, a natural problem is to describe the
geometry of $\iota(M)\equiv\iota$ through the extrinsic shape of simple test
curves in $M$. For example, choosing $M$-geodesics as test curves, one proves
that $\iota$ is totally geodesic if and only if the extrinsic shape of every
geodesic of $M$ is a geodesic of $Q$. Here, and in the rest of the paper, the
extrinsic shape of a curve $\gamma$ in $M$ is the curve $\iota\circ\gamma$.

Another fundamental result of this sort is the well-known theorem of Nomizu
and Yano [14], characterizing extrinsic spheres by the property that the
extrinsic shape of every circle in $M$ is a circle in $Q$. Recall that a
circle in a Riemannian manifold 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 $\gamma$ be a smooth unit-speed curve in $M$, and
let $\kappa$ be its geodesic curvature, i.e.,
$\kappa=\langle\mathop{}\\!\nabla_{\gamma^{\prime}}\gamma^{\prime},\mathop{}\\!\nabla_{\gamma^{\prime}}\gamma^{\prime}\rangle^{1/2}$,
where $\mathop{}\\!\nabla$ denotes the Levi-Civita connection of $M$. One says
that $\gamma$ has order two at the point $\gamma(s)$ if there exists a local
field of unit vectors $Y$ along $\gamma$ such that

$\begin{cases}\mathopen{}\mathclose{{}\left.\mathop{}\\!\nabla_{\gamma^{\prime}(t)}\gamma^{\prime}}\right\rvert_{t=s}=\kappa(s)Y(s),\\\
\mathopen{}\mathclose{{}\left.\mathop{}\\!\nabla_{\gamma^{\prime}(t)}Y}\right\rvert_{t=s}=-\kappa(s)\gamma^{\prime}(s).\end{cases}$

######  Theorem 1 ([1]).

The following statements are equivalent:

  1. (i)

$\iota$ 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.

  2. (ii)

For every orthonormal pair of vectors $u,v\in T_{p}M$, there exists a curve
$\gamma$, defined in a neighborhood of $0$, such that

    1. (a)

$\gamma(0)=p$, $\gamma^{\prime}(0)=u$, and
$\mathop{}\\!\nabla_{\gamma^{\prime}(0)}\gamma^{\prime}\rvert_{t=0}=\kappa(0)v$;

    2. (b)

The extrinsic shape $\iota\circ\gamma$ of $\gamma$ has order two at
$\iota(p)$;

    3. (c)

$\kappa^{\prime}(0)/\kappa(0)=\tilde{\kappa}^{\prime}(0)/\tilde{\kappa}(0)$,
where $\tilde{\kappa}$ is the geodesic curvature of $\iota\circ\gamma$.

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 submanifolds. 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 definition 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 curve
$\bar{\gamma}=\iota\circ\gamma$ lying on a surface is a pseudo-geodesic if the
acceleration vector $\bar{\gamma}^{\prime\prime}$ makes a constant angle
$\theta$ with the surface normal. Note that the angle is zero precisely when
$\gamma$ is a geodesic.

This definition extends straightforwardly to any Riemannian submanifold;
indeed, it is easy to see that the angle $\theta$ is constant if and only if
either $\gamma$ is a geodesic or else the ratio between the normal curvature
$\langle\bar{\gamma}^{\prime\prime},N\rangle$ and the geodesic curvature
$\kappa$ is constant (Lemma 13). Hence, for Wunderlich’s definition to make
sense in arbitrary dimension and codimension, one just needs to interpret the
normal curvature as
$\tau=\langle\alpha(\bar{\gamma}^{\prime},\bar{\gamma}^{\prime}),\alpha(\bar{\gamma}^{\prime},\bar{\gamma}^{\prime})\rangle^{1/2}$,
where $\alpha$ is the second fundamental form. In fact, we shall define
pseudo-geodesics 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 $p\in M$, a
tangent vector $v\in T_{p}M$, and a constant $c\neq 0$, the initial value
problem for the pseudo-geodesic equation $\kappa=c\tau$ is underdetermined.
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 remarks. 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 $c\in\mathbb{R}\setminus\\{0\\}$, the extrinsic shape of every
planar $c$-pseudo-geodesic of $(M,\iota^{\ast}\alpha)$ is planar, then $\iota$
is totally umbilic and, away from geodesic points, has parallel normalized
mean curvature vector. Conversely, if $\iota$ is totally umbilic, then the
extrinsic shape of every planar pseudo-geodesic of $(M,\iota^{\ast}\alpha)$ is
planar.

######  Corollary 3.

If $\iota$ is totally umbilic, then the extrinsic shape of every geodesic of
$M$ is planar.

######  Corollary 4 ([13, Theorem 2]).

If $\iota$ is a non-totally geodesic extrinsic sphere, then the extrinsic
shape of every geodesic of $M$ is a circle.

######  Remark 5.

If $\iota$ 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
preserve the planarity of additional curves without necessarily being totally
geodesic. Our next result answers this question negatively.

######  Proposition 6.

Suppose that $\iota$ 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
submanifolds 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 $\iota$ is totally umbilic. In particular, if
the extrinsic shape of every geodesic of $M$ is a circle, then $\iota$ is a
non-totally geodesic extrinsic sphere.

The paper is organized as follows. The next section presents some
preliminaries, 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. In section 4 we then define pseudo-geodesic;
in particular, by restricting our attention 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 $M\subset Q$ an immersed
submanifold. Identifying, as customary, the tangent space of $M$ at $p$ with
its image under the differential of the inclusion $M\hookrightarrow Q$, we
have the orthogonal decomposition

$T_{p}Q=T_{p}M\oplus N_{p}M,$

where $N_{p}M$ 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 $\mathfrak{X}(M)$ and $\bar{\mathfrak{X}}(M)$ denote the sets of smooth
vector fields _on_ and _along_ $M$, respectively. Let
$\mathfrak{X}(M)^{\perp}$ be the set of _normal_ vector fields along $M$.
Clearly, $\bar{\mathfrak{X}}(M)=\mathfrak{X}(M)\oplus\mathfrak{X}(M)^{\perp}$.
If $X\in\mathfrak{X}(M)$ and $\bar{X}\in\bar{\mathfrak{X}}(M)$, then

$\mathop{}\\!\widetilde{\nabla}_{X}\bar{X}=\mathop{}\\!\pi^{\top}\mathop{}\\!\widetilde{\nabla}_{X}\bar{X}+\mathop{}\\!\pi^{\perp}\mathop{}\\!\widetilde{\nabla}_{X}\bar{X},$

where $\mathop{}\\!\widetilde{\nabla}$ is the Levi-Civita connection of $Q$,
$\mathop{}\\!\pi^{\top}$ and $\mathop{}\\!\pi^{\perp}$ are the orthogonal
projections onto the tangent and normal bundle of $M$, and where both $X$ and
$\bar{X}$ are extended arbitrarily to $Q$.

In particular, if $\bar{X}=Y\in\mathfrak{X}(M)$, then

$\mathop{}\\!\widetilde{\nabla}_{X}Y=\mathop{}\\!\nabla_{X}Y+\alpha(X,Y).$

Here $\mathop{}\\!\nabla$ is the Levi-Civita connection of
$(M,\iota^{\ast}\tilde{g})$ and $\alpha$ the second fundamental form.

Similarly, if $N\in\mathfrak{X}(M)^{\perp}$, then, denoting by $A_{N}$ the
shape operator of $M$ with respect to $N$ and by $\mathop{}\\!\nabla^{\perp}$
the normal connection of $M$,

$\mathop{}\\!\widetilde{\nabla}_{X}N=A_{N}(X)+\mathop{}\\!\nabla^{\perp}_{X}N.$

The normal connection allows us to introduce a natural covariant
differentiation $\nabla^{\ast}$ for the second fundamental 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 $T_{p}M\times T_{p}M \to
N_{p}M$. For any smooth section $B$ of $F$ and any $X\in\mathfrak{X}(M)$, let
$\nabla^{\ast}_{X}B$ be the smooth section of $F$ given by

$\mathopen{}\mathclose{{}\left(\nabla_{X}^{\ast}B}\right)(Y,Z)=\mathop{}\\!\nabla^{\perp}_{X}(B(Y,Z))-B(\mathop{}\\!\nabla_{X}Y,Z)-B(Y,\mathop{}\\!\nabla_{X}Z).$

It is standard to prove that $\nabla^{\ast}$ is a connection in $F$.

We next turn our attention to totally umbilic submanifolds.

Given any normal vector $\eta\in NM$, we say that $M$ is umbilic in direction
$\eta$ if the shape operator $A_{\eta}$ 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 $Q$.

One can show that $M$ is totally umbilic if and only if, for every pair of
vector fields $X,Y\in\mathfrak{X}(M)$, the following relation holds between
the second fundamental form and the mean curvature vector $H$ of $M$:

$\alpha(X,Y)=\langle X,Y\rangle H.$

Recall that the mean curvature vector of $M$ is the normal vector field along
$M$ given by

$H=m^{-1}\operatorname{tr}(\alpha),$

where $m=\dim M$ and $\mathrm{tr}(\alpha)$ is the trace of $\alpha$.
Equivalently, in terms of a local orthonormal frame $(X_{1},\dotsc,X_{m})$ for
$TM$,

$H=m^{-1}\sum_{j=1}^{m}\alpha(X_{j},X_{j}).$

Among totally umbilic submanifolds, extrinsic spheres are particularly
important. 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 $\mathop{}\\!\nabla^{\perp}_{X}H=0$ for all $X\in\mathfrak{X}(M)$.
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 $H/\lVert H\rvert$ is well-defined. One says that $M$ has parallel
normalized mean curvature vector if $\nabla^{\perp}_{X}(H/\lVert H\rvert)=0$
for all $X\in\mathfrak{X}(M)$.

##  3\. Planar curves

In this section we define planar curves in a Riemannian manifold
$M\equiv(M,\langle\cdot{,}\cdot\rangle)$ and extend several well-known results
concerning circles.

######  Definition 8.

Let $\gamma\colon I\to M$ be a smooth, unit-speed curve, and denote by $T$ its
tangent vector. We say that $\gamma$ is planar if there exists a unit vector
field $Y$ along $\gamma$ and a smooth function $\kappa\colon I\to\mathbb{R}$
such that

(1) $\begin{cases}\mathop{}\\!\nabla_{T}T=\kappa Y,\\\
\mathop{}\\!\nabla_{T}Y=-\kappa T.\end{cases}$

######  Remarks.

  * –

In dimension two every regular curve is planar.

  * –

A geodesic is a planar curve with $\kappa=0$.

  * –

A planar curve with constant $\kappa>0$ 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

$\mathop{}\\!\nabla^{2}_{T}T+\kappa^{2}T=0,$

where
$\kappa=\langle\mathop{}\\!\nabla_{T}T,\mathop{}\\!\nabla_{T}T\rangle^{1/2}$
is the geodesic curvature. In the case of planar curves, the following lemma
holds.

######  Lemma 9 ([8, Lemma 2.3]).

A curve $\gamma$ is planar if and only if it satisfies the differential
equation

(2)
$\kappa\mathop{}\\!\nabla^{2}_{T}T+\kappa^{3}T-\kappa^{\prime}\mathop{}\\!\nabla_{T}T=0.$

###### Proof.

Notice that the lemma holds when $\gamma$ is a geodesic. Hence, we may assume
that $\kappa(t)>0$ for all $t\in I$.

If $\gamma$ is planar, then

$\displaystyle\mathop{}\\!\nabla^{2}_{T}T=\mathop{}\\!\nabla_{T}(\kappa Y)$
$\displaystyle=\kappa\mathop{}\\!\nabla_{T}Y+T(\kappa)Y$
$\displaystyle=-\kappa^{2}T+\kappa^{\prime}Y$
$\displaystyle=-\kappa^{2}T+\kappa^{\prime}\frac{\mathop{}\\!\nabla_{T}T}{\kappa},$

which shows that equation (2) holds.

Conversely, consider an arbitrary unit-speed curve $\gamma$, and define a unit
vector field $Y$ along $\gamma$ by

$Y=\frac{\mathop{}\\!\nabla_{T}T}{\kappa}.$

It follows that $\mathop{}\\!\nabla_{T}T=\kappa Y$ and

$\mathop{}\\!\nabla_{T}Y=\mathop{}\\!\nabla_{T}\mathopen{}\mathclose{{}\left(\frac{\mathop{}\\!\nabla_{T}T}{\kappa}}\right)=\frac{\mathop{}\\!\nabla^{2}_{T}T}{\kappa}+\mathopen{}\mathclose{{}\left(\frac{1}{\kappa}}\right)^{\prime}\mathop{}\\!\nabla_{T}T.$

In particular, if (2) holds, then

$\displaystyle\mathop{}\\!\nabla_{T}Y$
$\displaystyle=\frac{\frac{\kappa^{\prime}}{\kappa}\mathop{}\\!\nabla_{T}T-\kappa^{2}T}{\kappa}+\mathopen{}\mathclose{{}\left(\frac{1}{\kappa}}\right)^{\prime}\mathop{}\\!\nabla_{T}T$
$\displaystyle=-\kappa T,$

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 $p=\gamma(u)$ be an arbitrary point in the image of $\gamma$. The Cartan
development of $\gamma$ in the tangent space $T_{p}M$ is the unique curve
$\gamma^{\ast} \colon I\to T_{p}M$ such that

  1. (i)

$(\gamma^{\ast})^{\prime}(u)=T(u)$;

  2. (ii)

for all $t\in I$, the vector $(\gamma^{\ast})^{\prime}(t)$ is parallel—in the
Euclidean sense—to the parallel transport $\tau_{u}^{t}(T(t))$ of $T(t)$ from
$\gamma(t)$ to $\gamma(u)$ along $\gamma$.

######  Proposition 11.

A curve $\gamma$ is planar if and only if its development $\gamma^{\ast}$ in
the tangent space $T_{p}M$ is a regular planar curve in the ordinary Euclidean
sense.

###### Proof.

Assume that $\gamma$ is planar so that (1) holds. Let

$T^{\ast}(t)=\tau_{u}^{t}\,T(t)\quad\text{and}\quad
Y^{\ast}(t)=\tau_{u}^{t}\,Y(t).$

Since the map $\tau_{u}^{t}\colon T_{\gamma(u)}M\to T_{\gamma(t)}M$ is linear
and $\tau_{u}^{t+h}=\tau_{u}^{t}\circ\tau_{t}^{t+h}$, we obtain

$\displaystyle\mathopen{}\mathclose{{}\left(T^{\ast}}\right)^{\prime}(t)$
$\displaystyle=\lim_{h\to 0}\frac{T^{\ast}(t+h)-T^{\ast}(t)}{h}=\lim_{h\to
0}\frac{\tau_{u}^{t}\mathopen{}\mathclose{{}\left(\tau_{t}^{t+h}\,T(t+h)-T(t)}\right)}{h}$
$\displaystyle=\tau_{u}^{t}\,\lim_{h\to
0}\frac{\tau_{t}^{t+h}\,T(t+h)-T(t)}{h}$
$\displaystyle=\tau_{u}^{t}\,\nabla_{T}T(t).$

Likewise, we have

$\mathopen{}\mathclose{{}\left(Y^{\ast}}\right)^{\prime}(t)=\tau_{u}^{t}\,\nabla_{T}Y(t).$

By virtue of these identities, (1) implies

(3) $\mathopen{}\mathclose{{}\left(T^{\ast}}\right)^{\prime}=\kappa
Y^{\ast}\quad\text{and}\quad\mathopen{}\mathclose{{}\left(Y^{\ast}}\right)^{\prime}=-\kappa
T^{\ast}.$

Clearly, these equations express the fact that $\gamma^{\ast}$ is a planar
curve in $T_{p}M$.

Conversely, assume that the development $\gamma^{\ast}$ of $\gamma$ in
$T_{p}M$ is a regular planar curve. Then there exists a unit vector field
$Y^{\ast}$ along $\gamma^{\ast}$ and a smooth function $\kappa$ which,
together with $T^{\ast}=(\gamma^{\ast})^{\prime}$, satisfy (3). Since
$\tau_{u}^{t}$ is an isomorphism between $T_{\gamma(t)}M$ and
$T_{\gamma(u)}M$, 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 $x,y\in
T_{p}M$ and any smooth function $\kappa\colon\mathbb{R}\to\mathbb{R}$, there
exists a unique planar curve $\gamma\colon\mathbb{R}\to M$ satisfying (1) and
such that $\gamma(0)=p$, $T(0)=x$, and $Y(0)=y$.

###### Proof.

By the standard theory of ordinary differential equations, the problem defined
by the system (3) and the initial condition

$\gamma^{\ast}(0)=p,\quad T^{\ast}(0)=x,\quad Y^{\ast}(0)=y,$

has unique global solution $\gamma^{\ast}$ in $T_{p}M$. Since $M$ is complete,
it follows from [7, p. 172, Theorem 4.1] that there is a curve in $M$ whose
development in $T_{p}M$ is $\gamma^{\ast}$. By the proof of Proposition 11,
this is precisely the desired curve $\gamma$. ∎

##  4\. Planar pseudo-geodesics

In 1950 Wunderlich considered a natural extrinsic generalization of a geodesic
of a surface $\iota\colon S\hookrightarrow\mathbb{R}^{3}$. Noting that a curve
$\gamma$ is a geodesic of $S$ if and only if the ambient acceleration
$\bar{\gamma}^{\prime\prime}=(\iota\circ\gamma)^{\prime\prime}$ is parallel to
the surface unit normal $N$, he called a curve $\bar{\gamma}$ in $\iota(S)$ a
pseudo-geodesic if the angle $\theta$ between $\bar{\gamma}^{\prime\prime}$
and $N$ is constant.

The next lemma characterizes pseudo-geodesics in terms of curvature.

######  Lemma 13.

A curve $\bar{\gamma}$ in $\iota(S)$ is a pseudo-geodesic if and only if, for
some constant $c\in\mathbb{R}$, the geodesic curvature $\kappa$ and the normal
curvature $\tau$ satisfy $\kappa=c\tau$.

###### Proof.

Since a geodesic satisfies the lemma with $c=0$, we assume $\kappa\neq 0$. It
follows that

$\langle\bar{\gamma}^{\prime\prime},N\rangle=\lVert\bar{\gamma}^{\prime\prime}\rVert\cos(\theta)$

implies

$\displaystyle\cos(\theta)^{2}$
$\displaystyle=\frac{\tau^{2}}{\kappa^{2}+\tau^{2}},$
$\displaystyle\sin(\theta)^{2}$
$\displaystyle=\frac{\kappa^{2}}{\kappa^{2}+\tau^{2}}=\frac{1}{1+\mathopen{}\mathclose{{}\left(\tau/\kappa}\right)^{2}}.$

Note that $\theta$ is constant if and only if $\sin(\theta)^{2}$ is. Hence, if
$\kappa\neq 0$, then $\bar{\gamma}$ is a pseudo-geodesic precisely when the
ratio $\tau/\kappa$ is constant. ∎

Basing on this result, we extend Wunderlich’s definition as follows.

######  Definition 14.

Let $M$ be a Riemannian manifold, let $\varSigma$ be a smooth vector bundle of
rank $n$ over $M$, and let $\sigma$ be a smooth field of symmetric bilinear
forms $T_{p}M\times T_{p}M\to\varSigma_{p}$ on $M$. A unit-speed curve
$\gamma\colon I\to M$ is a ($c$-)pseudo-geodesic of $(M,\sigma)$ if there
exists a unit vector field $Y$ along $\gamma$ and a constant $c\in\mathbb{R}$
such that

$\displaystyle\mathop{}\\!\nabla_{T}T=\begin{cases}c\sigma(T,T)Y\quad&\text{if
}n=1;\\\ c\mathopen{}\mathclose{{}\left\lVert\sigma(T,T)}\right\rVert
Y\quad&\text{if }n\geq 2.\end{cases}$

Now, a fundamental property of geodesics is that, for any tangent vector $x\in
T_{p}M$, there exists a geodesic, defined for $\lvert t\rvert<\epsilon$ for
some $\epsilon>0$, such that $\gamma_{x}(0)=p$ and $\gamma_{x}^{\prime}(0)=x$.
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 $p\in M$. For any orthonormal pair of vectors $x,y$ in $T_{p}M$ and for
any constant $c\in\mathbb{R}$, there exists a planar pseudo-geodesic
$\gamma_{x,y}$ of $(M,\sigma)$, defined for $\lvert t\rvert<\epsilon$ for some
$\epsilon>0$, such that

$\gamma_{x,y}(0)=p,\quad
T_{x,y}(0)=x,\quad\nabla_{T_{x,y}}T_{x,y}(0)=c\sigma(x,x)y,$

where $T_{x,y}=\gamma_{x,y}^{\prime}$.

###### Proof.

We first consider the case where $n=1$, i.e., where $\sigma$ is a symmetric
two-tensor field on $M$.

Let $\gamma\colon I\to M$ be an arbitrary unit-speed curve in $M$. Let $Y$ be
an arbitrary unit vector field along $\gamma$. Then, by (1), it is clear that
$\gamma$ is a planar pseudo-geodesic of $(M,\sigma)$ if and only if there
exists $c\in\mathbb{R}$ such that

$\displaystyle\nabla_{T}T$ $\displaystyle= c\sigma(T,T)Y,$
$\displaystyle\nabla_{T}Y$ $\displaystyle= -c\sigma(T,T)T.$

We shall explore how these equations look in coordinates.

Suppose, thus, that $\gamma$ is contained in the domain of a smooth chart
$(u^{1},\dotsc,u^{m})$ for $M$ around $p$. Expanding $T$ and $Y$ in terms of
the coordinate frame $(\partial_{1}=\partial/\partial
u^{1},\dotsc,\partial_{m}=\partial/\partial u^{m})$, we obtain

$\displaystyle T$ $\displaystyle=T^{j}\partial_{j},$ $\displaystyle Y$
$\displaystyle=Y^{j}\partial_{j}.$

Using [9, Proposition 4.6], we compute

$\displaystyle\mathop{}\\!\nabla_{T}T$
$\displaystyle=\dot{T}^{k}\partial_{k}+T^{i}T^{j}\varGamma_{ij}^{k}\partial_{k},$
$\displaystyle\mathop{}\\!\nabla_{T}Y$
$\displaystyle=\dot{Y}^{k}\partial_{k}+T^{i}Y^{j}\varGamma_{ij}^{k}\partial_{k},$

where a dot indicates differentiation with respect to $t$ and
$\varGamma_{ij}^{k}$ is assumed to be evaluated along $\gamma$.

Likewise, expressing $\sigma$ in terms of the coordinate coframe
$(du^{1},\dotsc,du^{m})$, we get

$\sigma=\sigma_{ij}\mathop{}\\!du^{i}\otimes\mathop{}\\!du^{j}.$

It follows that

$\sigma(T(t),T(t))=\sigma_{ij}(\gamma(t))T^{i}(t)T^{j}(t).$

Summing up, $\gamma$ is a planar pseudo-geodesic of $(M,\sigma)$ if and only
if, for some $c\in\mathbb{R}$ and every $k=1,\dotsc,m$, the following two
equations hold:

(4) $\displaystyle\dot{T}^{k}$
$\displaystyle=c\sigma_{ij}T^{i}T^{j}Y^{k}-\varGamma_{ij}^{k}T^{i}T^{j},$ (5)
$\displaystyle\dot{Y}^{k}$
$\displaystyle=\varGamma_{ij}^{k}T^{i}Y^{j}-c\sigma_{ij}T^{i}T^{j}T^{k}.$

Together with $\dot{u}^{k}=T^{k}$, equations (4) and (5) define a system of
$3m$ ordinary differential equations in the $3m$ unknown functions
$(u^{k},T^{k},Y^{k})_{k=1}^{m}$, which admits a unique, maximal local solution
for any initial condition $(u^{k}(0),T^{k}(0),Y^{k}(0))_{k=1}^{m}$.

We next examine the case where $n\geq 2$. Let $(E_{1},\dotsc,E_{n})$ be a
smooth orthonormal frame for $\varSigma$ along $\gamma$. Then there are
symmetric two-tensors
$\sigma^{1}=\langle\sigma,N_{1}\rangle,\dotsc,\sigma^{n}=\langle\sigma,N_{n}\rangle$
on $M$, and

$\sigma=\sigma^{1}N_{1}+\dotsb\sigma^{n}N_{n}.$

It follows that

$\displaystyle\mathopen{}\mathclose{{}\left\lVert\sigma(T,T)}\right\rVert$
$\displaystyle=\mathopen{}\mathclose{{}\left<\sigma(T,T),\sigma(T,T)}\right>^{1/2}$
$\displaystyle=\mathopen{}\mathclose{{}\left<\sigma^{s}(T,T)N_{s},\sigma^{s}(T,T)N_{s}}\right>^{1/2}$
$\displaystyle=\mathopen{}\mathclose{{}\left<\sigma^{s}_{ij}T^{i}T^{j}N_{s},\sigma^{s}_{ij}T^{i}T^{j}N_{s}}\right>^{1/2}$
$\displaystyle=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(\sigma^{1}_{ij}T^{i}T^{j}}\right)^{2}+\dotsb+\mathopen{}\mathclose{{}\left(\sigma^{n}_{ij}T^{i}T^{j}}\right)^{2}}\right)^{1/2},$

and so equations (4) and (5) become

(6) $\displaystyle\dot{T}^{k}$
$\displaystyle=c\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(\sigma^{1}_{ij}T^{i}T^{j}}\right)^{2}+\dotsb+\mathopen{}\mathclose{{}\left(\sigma^{n}_{ij}T^{i}T^{j}}\right)^{2}}\right)^{1/2}Y^{k}-\varGamma_{ij}^{k}T^{i}T^{j},$
(7) $\displaystyle\dot{Y}^{k}$
$\displaystyle=\varGamma_{ij}^{k}T^{i}Y^{j}-c\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(\sigma^{1}_{ij}T^{i}T^{j}}\right)^{2}+\dotsb+\mathopen{}\mathclose{{}\left(\sigma^{n}_{ij}T^{i}T^{j}}\right)^{2}}\right)^{1/2}T^{k}.$

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.

######  Lemma 16.

Suppose that, for each orthonormal pair of tangent vectors $x,y$ in $T_{p}M$,
either $\alpha(x,y)=0$ or $\alpha(x,x)=\alpha(y,y)=0$. Then the following
conclusions hold:

  1. (i)

$\alpha(x,x)=\pm\alpha(y,y)$ for any orthonormal $x,y$ in $T_{p}M$.

  2. (ii)

If $\alpha(x,x)=0$ for some $x$ in $T_{p}M$, then $\alpha$ 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 $\alpha(x+y,x-y)=0$ or else
$\alpha(x+y,x+y)=\alpha(x-y,x-y)=0$. It is easy to check, using bilinearity
and symmetry, that the first condition implies $\alpha(x,x)=\alpha(y,y)$,
whereas the second $\alpha(x,x)=-\alpha(y,y)$.

Now we prove (ii). Suppose there is a unit vector $x_{1}$ in $T_{p}M$ such
that $\alpha(x_{1},x_{1})=0$. Then, if $(x_{1},\dotsc,x_{m})$ is an
orthonormal basis of $T_{p}M$, it follows from (i) that
$\alpha(x_{j},x_{j})=0$ for all $j=1,\dotsc,m$. In fact,
$\alpha(x_{j}+x_{k},x_{j}+x_{k})=0$ for all $j,k=1,\dotsc,m$, because
$2^{-1/2}(x_{j}+x_{k})$ and $x_{h}$ are orthonormal when $h\neq j,k$. Since

$\alpha(x_{j}+x_{k},x_{j}+x_{k})=2\alpha(x_{j},x_{k}),$

we conclude that $\alpha(x_{j},x_{k})=0$ for all $j$ and $k$. Hence, by
bilinearity, $\alpha$ vanishes at $p$. ∎

###### Proof of Theorem 2.

By Lemma 9, the extrinsic shape of $\gamma$ is a planar curve in $Q$ precisely
when

(8)
$\tilde{\kappa}\mathop{}\\!\widetilde{\nabla}^{2}_{T}T+\tilde{\kappa}^{3}T-\tilde{\kappa}^{\prime}\mathop{}\\!\widetilde{\nabla}_{T}T=0,$

where
$\tilde{\kappa}=\langle\mathop{}\\!\widetilde{\nabla}_{T}T,\mathop{}\\!\widetilde{\nabla}_{T}T\rangle^{1/2}$,
and where we identified $\widetilde{T}=(\iota\circ\gamma)^{\prime}$ with $T$.
Since
$\mathop{}\\!\widetilde{\nabla}_{T}T=\mathop{}\\!\nabla_{T}T+\alpha(T,T)$,
denoting by $\tau$ the length of $\alpha(T,T)$, it follows that
$\tilde{\kappa}=\sqrt{\kappa^{2}+\tau^{2}}$.

Let $p\in M$. 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 $\gamma$ such that $\gamma(0)=p$ and $\gamma^{\prime}(0)=x$ there
exists an open interval $(-\epsilon,\epsilon)$ such that $\tau(t)\neq 0$ in
$(-\epsilon,\epsilon)$.

Assume that $x$ is not asymptotic. Then, in $(-\epsilon,\epsilon)$,

$\tilde{\kappa}^{\prime}=\frac{2\kappa\kappa^{\prime}+2\tau\tau^{\prime}}{2\tilde{\kappa}}=\frac{\tilde{\kappa}\mathopen{}\mathclose{{}\left(\kappa\kappa^{\prime}+\tau\tau^{\prime}}\right)}{\kappa^{2}+\tau^{2}},$

so that equation (8) becomes

(9)
$\mathop{}\\!\widetilde{\nabla}^{2}_{T}T+\tilde{\kappa}^{2}T-\frac{\kappa\kappa^{\prime}+\tau\tau^{\prime}}{\kappa^{2}+\tau^{2}}\mathop{}\\!\widetilde{\nabla}_{T}T=0.$

Moreover, by computing

$\displaystyle\mathop{}\\!\widetilde{\nabla}_{T}T$
$\displaystyle=\mathop{}\\!\nabla_{T}T+\alpha(T,T),$
$\displaystyle\mathop{}\\!\widetilde{\nabla}^{2}_{T}T$
$\displaystyle=\mathop{}\\!\nabla^{2}_{T}T+\alpha(T,\mathop{}\\!\nabla_{T}T)+\mathop{}\\!\widetilde{\nabla}_{T}\alpha(T,T),$

we see that (9) is equivalent to

$\mathop{}\\!\nabla^{2}_{T}T+\alpha(T,\mathop{}\\!\nabla_{T}T)+\mathop{}\\!\widetilde{\nabla}_{T}\alpha(T,T)+\tilde{\kappa}^{2}T-\frac{\kappa\kappa^{\prime}+\tau\tau^{\prime}}{\kappa^{2}+\tau^{2}}\mathopen{}\mathclose{{}\left(\mathop{}\\!\nabla_{T}T+\alpha(T,T)}\right)=0.$

Decomposing into tangent and normal components, we finally obtain

(10)
$\displaystyle\mathop{}\\!\nabla^{2}_{T}T+A_{\alpha(T,T)}T+\mathopen{}\mathclose{{}\left(\kappa^{2}+\tau^{2}}\right)T-\frac{\kappa\kappa^{\prime}+\tau\tau^{\prime}}{\kappa^{2}+\tau^{2}}\mathop{}\\!\nabla_{T}T=0,$
(11)
$\displaystyle\alpha(T,\mathop{}\\!\nabla_{T}T)+\mathop{}\\!\nabla^{\perp}_{T}\alpha(T,T)-\frac{\kappa\kappa^{\prime}+\tau\tau^{\prime}}{\kappa^{2}+\tau^{2}}\alpha(T,T)=0.$

In particular, if $\kappa=c\tau$ for some $c\neq 0$ and $\gamma$ is planar,
then (10) and (11) simplify to

$\displaystyle A_{\alpha(T,T)}T+\tau^{2}T=0,$ (12)
$\displaystyle\alpha(T,\mathop{}\\!\nabla_{T}T)+\mathop{}\\!\nabla^{\perp}_{T}\alpha(T,T)-\frac{\tau^{\prime}}{\tau}\alpha(T,T)=0.$

Using

$\mathopen{}\mathclose{{}\left(\nabla_{T}^{\ast}\alpha}\right)(T,T)=\mathop{}\\!\nabla^{\perp}_{T}\alpha(T,T)-2\alpha(\mathop{}\\!\nabla_{T}T,T),$

we rewrite (12) as

$3\alpha(T,\mathop{}\\!\nabla_{T}T)+\mathopen{}\mathclose{{}\left(\nabla_{T}^{\ast}\alpha}\right)(T,T)-\frac{\tau^{\prime}}{\tau}\alpha(T,T)=0.$

At $t=0$, since $(\mathop{}\\!\nabla_{T}T)_{t=0}=c\tau(0)Y(0)$, the last
equation specializes to

$\alpha(T(0),Y(0))=\frac{1}{3c\tau(0)}\mathopen{}\mathclose{{}\left(\frac{\tau^{\prime}(0)}{\tau(0)}\alpha(T(0),T(0))-\mathopen{}\mathclose{{}\left(\nabla_{T}^{\ast}\alpha}\right)(T(0),T(0))}\right).$

This equation implies that, given a unit vector $x\in T_{p}M$ that is not
asymptotic, the value $\alpha(x,y)$ does not depend on $y\in T_{p}M$, so long
as $\langle x,y\rangle=0$. In fact, since $\alpha(x,-y)=-\alpha(x,y)$, it is
clear that $\alpha(x,y)=0$ for every $x$ and $y$ such that $\langle
x,y\rangle=0$.

If, on the other hand, $x$ is asymptotic, then, for each $y$ in the orthogonal
complement of $x$ in $T_{p}M$, either $\alpha(x,y)=0$ or $\alpha(y,y)=0$;
indeed, if $\alpha(y,y)\neq 0$, then $\alpha(y,x)=0$ by the previous argument.

Summing up, we have shown that, if (12) holds for every nonasymptotic $x\in
T_{p}M$, then, for each orthogonal pair of vectors $x,y$ in $T_{p}M$, either
$\alpha(x,y)=0$ or $\alpha(x,x)=\alpha(y,y)=0$. Hence, by Lemma 16, if there
is $x\in T_{p}M$ such that $\alpha(x,x)=0$, then $p$ is a geodesic point.

Assume that there is no such vector. It follows that there exists a
neighbourhood $U$ 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
$\overline{\alpha}=\alpha(T,T)/\tau$ along $\iota\circ\gamma$. Equation (12)
therefore simplifies to

$\mathop{}\\!\nabla^{\perp}_{T}\alpha(T,T)-\frac{\tau^{\prime}}{\tau}\alpha(T,T)=0.$

Substituting $\alpha(T,T)=\tau\overline{\alpha}$, this becomes

$\mathop{}\\!\nabla^{\perp}_{T}\overline{\alpha}=0,$

as desired.

Conversely, suppose that $\iota$ is totally umbilic. Note that, if $\gamma$ is
a pseudo-geodesic, then equation (8) gives an identity whenever $\gamma(t)$ is
a geodesic point. Hence we may assume that the mean curvature vector never
vanishes. It follows that $\alpha(T,\mathop{}\\!\nabla_{T}T)=0$ and
$\overline{\alpha}=\alpha(T,T)/\tau$ coincides with the normalized mean
curvature vector along $\iota\circ\gamma$. Moreover,

$\mathop{}\\!\pi^{\top}\mathop{}\\!\widetilde{\nabla}_{T}\overline{\alpha}=\langle\mathop{}\\!\widetilde{\nabla}_{T}\overline{\alpha},T\rangle
T=-\langle\overline{\alpha},\mathop{}\\!\widetilde{\nabla}_{T}T\rangle=-\tau
T.$

A straightforward computation would reveal that equations (10) and (11) are
now equivalent to

$\displaystyle\mathop{}\\!\nabla^{2}_{T}T+\kappa^{2}T-\frac{\kappa\kappa^{\prime}+\tau\tau^{\prime}}{\kappa^{2}+\tau^{2}}\mathop{}\\!\nabla_{T}T=0,$
(13)
$\displaystyle\tau\mathop{}\\!\nabla^{\perp}_{T}\overline{\alpha}+\tau^{\prime}\overline{\alpha}-\tau\frac{\kappa\kappa^{\prime}+\tau\tau^{\prime}}{\kappa^{2}+\tau^{2}}\overline{\alpha}=0.$

Suppose that $\gamma$ is a $c$-pseudo-geodesic. If $c=0$, then $\gamma$ is a
geodesic. In that case the first equation gives an identity whereas the second
reduces to $\nabla^{\perp}_{T}\overline{\alpha}=0$. On the other hand, if
$c\neq 0$, then substituting $\tau=\kappa/c$ in the first and $\kappa=c\tau$
in the second gives

$\displaystyle\kappa\mathop{}\\!\nabla^{2}_{T}T+\kappa^{3}T-\kappa^{\prime}\mathop{}\\!\nabla_{T}T=0,$
$\displaystyle\mathop{}\\!\nabla^{\perp}_{T}\overline{\alpha}=0.$

Evidently, these two are fulfilled exactly when $\gamma$ is planar and the
normalized mean curvature vector $\overline{\alpha}$ is parallel. ∎

###### Proof of Proposition 6.

Assume the hypothesis of the proposition. Then, for
$\nabla^{\perp}_{T}\overline{\alpha}=0$, equation (13) is equivalent to

$\kappa\mathopen{}\mathclose{{}\left(\tau^{\prime}\kappa-\tau\kappa^{\prime}}\right)=0.$

Assume that $\iota\circ\gamma$ is planar. Then, on the open subset where
$\kappa(t)\neq 0$,

$\mathopen{}\mathclose{{}\left(\frac{\tau}{\kappa}}\right)^{\prime}=0,$

which implies that $\gamma$ 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 $\langle\alpha(\cdot{,}\cdot),N\rangle$. Clearly, if
$\gamma$ is a geodesic, then $\tilde{\kappa}=\tau$ and

$\mathop{}\\!\widetilde{\nabla}_{T}T=\alpha(T,T),$

so that equation (8) reads

(14)
$\tau\mathop{}\\!\widetilde{\nabla}_{T}\alpha(T,T)+\tau^{3}T-\tau^{\prime}\alpha(T,T)=0.$

In particular, if $\tau$ is strictly positive, then $\alpha(T,T)=\pm\tau N$,
and therefore (14) becomes

(15) $\pm\tau^{2}A(T)+\tau^{3}T=0,$

being $A$ the shape operator.

Let $\mathbb{S}_{p}^{m-1}$ be the unit sphere in $T_{p}M$. 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 $x,y\in\mathbb{S}_{p}^{m-1}$ are linearly independent
and nonasymptotic, then $h(x)=\pm h(y)$. Indeed, if $h(x+y)\neq 0$, then

$h(x+y)(x+y)=A(x+y)=A(x)+A(y)=h(x)x+h(y)y,$

which implies $h(x)=h(y)$. On the contrary, if $x+y$ is asymptotic, then

$0=\langle A(x+y),x+y\rangle=(h(x)+h(y))(1+\langle x,y\rangle),$

implying $h(x)=-h(y)$. We conclude that $\mathbb{S}_{p}^{m-1}$ can be
decomposed as the union of the following subsets:

$\displaystyle\\{x\in\mathbb{S}_{p}^{m-1}\mid\langle A(x),x\rangle=0\\},$
$\displaystyle\\{x\in\mathbb{S}_{p}^{m-1}\mid\langle A(x),x\rangle=h(y)\\},$
$\displaystyle\\{x\in\mathbb{S}_{p}^{m-1}\mid\langle A(x),x\rangle=-h(y)\\}.$

It is clear that, since $x\mapsto\langle A(x),x\rangle$ is a continuous
function $\mathbb{S}_{p}^{m-1}\to\mathbb{R}$, only one of these sets can be
nonempty. This 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
codimension 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 $\iota\colon M\hookrightarrow Q$ be an isometric immersion. We say that
$\iota$ is isotropic at $p\in M$ if
$\langle\alpha(x,x),\alpha(x,x)\rangle=\lambda_{p}$ for all unit vectors $x\in
T_{p}M$. In particular, if $\iota$ is isotropic at every point $p\in M$, then
$\iota$ is called a totally isotropic immersion. A totally isotropic immersion
is constant isotropic if $\lambda_{p}$ is constant on $M$.

######  Theorem 18.

If the extrinsic shape of every geodesic of $M$ is planar, then $\iota$ is
totally isotropic. In particular, if the extrinsic shape of every geodesic of
$M$ is a circle, then $\iota$ 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 $\iota$ is a non-totally geodesic constant isotropic immersion exactly
when the extrinsic shape of every geodesic of $M$ is a circle and
$(\nabla_{X}^{\ast}\alpha)(X,X)=0$ for all $X\in\mathfrak{X}(M)$ [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 $\iota$ is constant isotropic, and thereby any such extrinsic shape is
either a geodesic or a circle; see [16].

###### Proof.

By [15, Lemma 1], $\iota$ is isotropic at $p$ exactly when
$\langle\alpha(x,x),\alpha(x,y)\rangle=0$ for every orthonormal pair of
vectors $x,y$ in $T_{p}M$. Obviously, if $\alpha(x,x)=0$, then
$\langle\alpha(x,x),\alpha(x,y)\rangle=0$ for every $y$, and so we may assume
that $x$ is not asymptotic.

Let $\gamma$ be the (unit-speed) geodesic originating from $p$ with tangent
vector $T(0)=x$. Let $Y$ be the parallel transport of $y$ along $\gamma$. Then

$\displaystyle\langle\alpha(x,x),\alpha(x,y)\rangle$
$\displaystyle=\langle\alpha(T,T),\alpha(T,Y)\rangle(0)$
$\displaystyle=\langle\alpha(T,T),\mathop{}\\!\widetilde{\nabla}_{T}Y\rangle(0)$
$\displaystyle=-\langle\mathop{}\\!\widetilde{\nabla}_{T}\alpha(T,T),Y\rangle(0).$

Here we have used the Gauss formula as well as orthogonality of $Y$ and
$\alpha(T,T)$.

We now show that
$\langle\mathop{}\\!\widetilde{\nabla}_{T}\alpha(T,T),Y\rangle=0$ if
$\iota\circ\gamma$ is planar. Indeed, since $\gamma$ is a geodesic, equation
(8) gives

(16)
$\tau\mathop{}\\!\widetilde{\nabla}_{T}\alpha(T,T)=-\tau^{3}T+\tau^{\prime}\alpha(T,T),$

which implies
$\langle\mathop{}\\!\widetilde{\nabla}_{T}\alpha(T,T),Y\rangle=0$. This proves
the first part of the theorem.

For the second part, assume that $\iota\circ\gamma$ is a circle, so that
$\tau>0$ and $\tau^{\prime}=0$. Then (16) implies

$-\langle\mathop{}\\!\widetilde{\nabla}_{T}\alpha(T,T),T\rangle=\langle\alpha(T,T),\mathop{}\\!\widetilde{\nabla}_{T}T\rangle=\langle\alpha(T,T),\alpha(T,T)\rangle=\tau^{2}.$

The last equality shows that $\langle\alpha(T,T),\alpha(T,T)\rangle$ is
constant along $\gamma$, and from here the statement follows easily. ∎

## 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 Kôzaki and Sadahiro Maeda, _A characterization of extrinsic spheres in a Riemannian manifold_ , Tsukuba J. Math. 26 (2002), no. 2, 291–297.
  * [9] John M. Lee, _Introduction to Riemannian manifolds_ , second ed., Graduate Texts in Mathematics, 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 Riemannian geometry_ , 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 Kähler immersions_ , Canadian J. Math. 17 (1965), 907–915.
  * [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 pseudogeodätische Linien zweier Kegel sind_ , Monatsh. Math. 54 (1950), 55–70.
  * [20] Walter Wunderlich, _Raumkurven, die pseudogeodätische Linien eines Zylinders und eines Kegels sind_ , Compositio Math. 8 (1950), 169–184.